| L(s) = 1 | − 4·4-s − 8·5-s + 10·16-s + 32·20-s + 36·25-s − 16·37-s − 48·41-s − 16·43-s + 48·47-s − 16·59-s − 20·64-s − 80·80-s − 16·83-s + 48·89-s − 144·100-s + 32·101-s − 16·109-s + 64·121-s − 120·125-s + 127-s + 131-s + 137-s + 139-s + 64·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·4-s − 3.57·5-s + 5/2·16-s + 7.15·20-s + 36/5·25-s − 2.63·37-s − 7.49·41-s − 2.43·43-s + 7.00·47-s − 2.08·59-s − 5/2·64-s − 8.94·80-s − 1.75·83-s + 5.08·89-s − 14.3·100-s + 3.18·101-s − 1.53·109-s + 5.81·121-s − 10.7·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.250044030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250044030\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{8} \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 64 T^{2} + 1952 T^{4} - 3392 p T^{6} + 489570 T^{8} - 3392 p^{3} T^{10} + 1952 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 16 T^{2} + 308 T^{4} - 5424 T^{6} + 50118 T^{8} - 5424 p^{2} T^{10} + 308 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 28 T^{2} - 72 T^{3} + 468 T^{4} - 72 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 112 T^{2} + 5956 T^{4} - 197328 T^{6} + 4474022 T^{8} - 197328 p^{2} T^{10} + 5956 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 + 48 T^{2} + 980 T^{4} - 4592 T^{6} - 398298 T^{8} - 4592 p^{2} T^{10} + 980 p^{4} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 - 144 T^{2} + 8384 T^{4} - 272784 T^{6} + 7329058 T^{8} - 272784 p^{2} T^{10} + 8384 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( 1 - 112 T^{2} + 5984 T^{4} - 201840 T^{6} + 5984898 T^{8} - 201840 p^{2} T^{10} + 5984 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 8 T + 132 T^{2} + 784 T^{3} + 7188 T^{4} + 784 p T^{5} + 132 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 24 T + 372 T^{2} + 3720 T^{3} + 28158 T^{4} + 3720 p T^{5} + 372 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 8 T + 76 T^{2} + 256 T^{3} + 1492 T^{4} + 256 p T^{5} + 76 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 24 T + 380 T^{2} - 3968 T^{3} + 31844 T^{4} - 3968 p T^{5} + 380 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 120 T^{2} + 14060 T^{4} - 973896 T^{6} + 61968070 T^{8} - 973896 p^{2} T^{10} + 14060 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 8 T + 176 T^{2} + 888 T^{3} + 12914 T^{4} + 888 p T^{5} + 176 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 304 T^{2} + 47620 T^{4} - 4871952 T^{6} + 350453414 T^{8} - 4871952 p^{2} T^{10} + 47620 p^{4} T^{12} - 304 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 164 T^{2} - 200 T^{3} + 14052 T^{4} - 200 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 352 T^{2} + 54788 T^{4} - 5305888 T^{6} + 402214086 T^{8} - 5305888 p^{2} T^{10} + 54788 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 184 T^{2} + 31436 T^{4} - 3096008 T^{6} + 278954470 T^{8} - 3096008 p^{2} T^{10} + 31436 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 8 T + 324 T^{2} + 1896 T^{3} + 40022 T^{4} + 1896 p T^{5} + 324 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 24 T + 428 T^{2} - 5032 T^{3} + 54470 T^{4} - 5032 p T^{5} + 428 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 504 T^{2} + 122652 T^{4} - 19352520 T^{6} + 2189485382 T^{8} - 19352520 p^{2} T^{10} + 122652 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.56261234531675009138162675529, −3.35043303929357529423074999409, −3.22929786521817187174538077070, −3.16897923865927614949982482506, −3.11672587573055180415575585472, −3.02783871320143849565770411778, −2.94725896217975531598869772464, −2.93231896218991279077269239094, −2.51272607548319134652902993215, −2.35695287863784164822966327030, −2.18711384297703847944924540450, −1.99654837043085509419831374717, −1.98870849990091482803563009914, −1.91157660460685744227134326656, −1.72990662149074463586122804628, −1.63155347064851898805546186024, −1.57746748823622228977191107025, −1.07357018845531955098886136309, −1.06253162820951206755680538890, −0.881932334721022930931055384892, −0.862218762955715603395576579062, −0.38844781359066976175666327225, −0.38008234216554860548016116041, −0.35176326154958902274885877961, −0.28803152707071084794015023944,
0.28803152707071084794015023944, 0.35176326154958902274885877961, 0.38008234216554860548016116041, 0.38844781359066976175666327225, 0.862218762955715603395576579062, 0.881932334721022930931055384892, 1.06253162820951206755680538890, 1.07357018845531955098886136309, 1.57746748823622228977191107025, 1.63155347064851898805546186024, 1.72990662149074463586122804628, 1.91157660460685744227134326656, 1.98870849990091482803563009914, 1.99654837043085509419831374717, 2.18711384297703847944924540450, 2.35695287863784164822966327030, 2.51272607548319134652902993215, 2.93231896218991279077269239094, 2.94725896217975531598869772464, 3.02783871320143849565770411778, 3.11672587573055180415575585472, 3.16897923865927614949982482506, 3.22929786521817187174538077070, 3.35043303929357529423074999409, 3.56261234531675009138162675529
Plot not available for L-functions of degree greater than 10.
