| L(s) = 1 | − 16·7-s + 4·9-s + 184·17-s + 16·23-s + 108·25-s + 768·31-s + 600·41-s + 32·47-s − 828·49-s − 64·63-s + 816·71-s − 824·73-s + 800·79-s − 678·81-s − 1.14e3·89-s + 4.40e3·97-s − 6.92e3·103-s + 392·113-s − 2.94e3·119-s + 996·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 736·153-s + ⋯ |
| L(s) = 1 | − 0.863·7-s + 4/27·9-s + 2.62·17-s + 0.145·23-s + 0.863·25-s + 4.44·31-s + 2.28·41-s + 0.0993·47-s − 2.41·49-s − 0.127·63-s + 1.36·71-s − 1.32·73-s + 1.13·79-s − 0.930·81-s − 1.36·89-s + 4.61·97-s − 6.62·103-s + 0.326·113-s − 2.26·119-s + 0.748·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.388·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.368159654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.368159654\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 694 T^{4} - 4 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 31094 T^{4} - 108 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 8 T + 510 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 996 T^{2} + 27974 p^{2} T^{4} - 996 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 332 T^{2} - 7598826 T^{4} - 332 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 92 T + 5030 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1196 p T^{2} + 223149174 T^{4} - 1196 p^{7} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 8798 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 84428 T^{2} + 2937799638 T^{4} - 84428 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 384 T + 84158 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 169004 T^{2} + 11993714550 T^{4} - 169004 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 300 T + 157270 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 270628 T^{2} + 30844604694 T^{4} - 270628 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 16 T + 114782 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 576620 T^{2} + 127450023606 T^{4} - 576620 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 509604 T^{2} + 127606046294 T^{4} - 509604 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 394892 T^{2} + 121971578838 T^{4} - 394892 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 526916 T^{2} + 240544281654 T^{4} - 526916 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 408 T + 672766 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 412 T + 690678 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 400 T + 963870 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1821572 T^{2} + 1476121597686 T^{4} - 1821572 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 572 T + 845846 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 2204 T + 2633478 T^{2} - 2204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.235845251851030795159415372678, −8.064819607428414646995240713303, −7.901307224953083046989586829980, −7.43251886800431269972583550192, −7.36070689651064354890277132681, −6.87113830797118259727467633018, −6.66230286552062815960847666733, −6.47694846727435404352580492499, −6.10409410486400240664974232508, −5.81959117671238354856040867616, −5.81084740674813120146070285562, −5.14066812895528069451341746572, −5.08165303807746959407653595815, −4.61411457740602034992541811191, −4.30436513088479533520626670424, −4.21474652664409196925636977490, −3.51139663308603471988417251481, −3.25943137238616784658199611740, −3.01124314633498597498321201195, −2.78254031657833997659732797494, −2.40611688742016800042345132177, −1.67046399666799519507505707816, −1.16498841298729587606749404801, −0.854739186018092801519651429586, −0.53076736711054078274571203394,
0.53076736711054078274571203394, 0.854739186018092801519651429586, 1.16498841298729587606749404801, 1.67046399666799519507505707816, 2.40611688742016800042345132177, 2.78254031657833997659732797494, 3.01124314633498597498321201195, 3.25943137238616784658199611740, 3.51139663308603471988417251481, 4.21474652664409196925636977490, 4.30436513088479533520626670424, 4.61411457740602034992541811191, 5.08165303807746959407653595815, 5.14066812895528069451341746572, 5.81084740674813120146070285562, 5.81959117671238354856040867616, 6.10409410486400240664974232508, 6.47694846727435404352580492499, 6.66230286552062815960847666733, 6.87113830797118259727467633018, 7.36070689651064354890277132681, 7.43251886800431269972583550192, 7.901307224953083046989586829980, 8.064819607428414646995240713303, 8.235845251851030795159415372678
