| L(s) = 1 | − 4·5-s − 6·9-s + 20·13-s + 2·25-s + 24·37-s + 24·41-s + 24·45-s − 6·49-s − 8·53-s − 80·65-s + 5·81-s + 64·89-s − 120·117-s + 34·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 102·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2·9-s + 5.54·13-s + 2/5·25-s + 3.94·37-s + 3.74·41-s + 3.57·45-s − 6/7·49-s − 1.09·53-s − 9.92·65-s + 5/9·81-s + 6.78·89-s − 11.0·117-s + 3.09·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 7.84·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.32999422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.32999422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 2 T + p T^{2} + 16 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( ( 1 + T^{2} )^{6} \) |
| good | 3 | \( ( 1 + p T^{2} + 11 T^{4} + 22 T^{6} + 11 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - 17 T^{2} + 339 T^{4} - 3302 T^{6} + 339 p^{2} T^{8} - 17 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 5 T + 37 T^{2} - 128 T^{3} + 37 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 17 | \( ( 1 - 67 T^{2} + 2107 T^{4} - 42562 T^{6} + 2107 p^{2} T^{8} - 67 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 86 T^{2} + 3451 T^{4} - 82508 T^{6} + 3451 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 2 p T^{2} + 1631 T^{4} - 47556 T^{6} + 1631 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 35 T^{2} + 2651 T^{4} - 58482 T^{6} + 2651 p^{2} T^{8} - 35 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + 126 T^{2} + 7871 T^{4} + 302596 T^{6} + 7871 p^{2} T^{8} + 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 6 T + 47 T^{2} - 412 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 2 T + p T^{2} )^{12} \) |
| 43 | \( ( 1 + 42 T^{2} + 2631 T^{4} + 184908 T^{6} + 2631 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 217 T^{2} + 21267 T^{4} - 1247782 T^{6} + 21267 p^{2} T^{8} - 217 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + 2 T + 119 T^{2} + 100 T^{3} + 119 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 59 | \( ( 1 - 226 T^{2} + 26123 T^{4} - 1881444 T^{6} + 26123 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 154 T^{2} + 16731 T^{4} - 1168852 T^{6} + 16731 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 + 62 T^{2} + 13191 T^{4} + 510788 T^{6} + 13191 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 230 T^{2} + 22559 T^{4} + 1599636 T^{6} + 22559 p^{2} T^{8} + 230 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 222 T^{2} + 28911 T^{4} - 2587908 T^{6} + 28911 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 143 T^{2} + 19635 T^{4} + 1513418 T^{6} + 19635 p^{2} T^{8} + 143 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 266 T^{2} + 41467 T^{4} + 4121204 T^{6} + 41467 p^{2} T^{8} + 266 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 16 T + 311 T^{2} - 2864 T^{3} + 311 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 97 | \( ( 1 - 195 T^{2} + 37595 T^{4} - 39106 p T^{6} + 37595 p^{2} T^{8} - 195 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.80586138128467381456020987468, −2.59674807405601397233540262213, −2.51188338649749123929196058510, −2.49344292771307493841326066102, −2.49166400752802777459498586444, −2.40391553191293171297846619487, −2.37447515356949653604174520290, −2.23109067361082671130569412224, −2.14542622047763174825906926865, −1.86949305910042269963600738626, −1.73878592638556919401688509093, −1.72493627193917410591681847926, −1.63683789689803803790375716731, −1.43558303572464063652786166770, −1.39836020346705741124725404928, −1.38061566956664894526737355602, −1.30596269004480717513504314574, −0.977309676380297827270965574519, −0.869425635117852436188195720937, −0.77484950596102852030747036810, −0.70965581210636465295987588997, −0.66345743022687287194162179831, −0.51679224972311058651041806847, −0.43929898272620759375999701245, −0.13842343463171370599553334218,
0.13842343463171370599553334218, 0.43929898272620759375999701245, 0.51679224972311058651041806847, 0.66345743022687287194162179831, 0.70965581210636465295987588997, 0.77484950596102852030747036810, 0.869425635117852436188195720937, 0.977309676380297827270965574519, 1.30596269004480717513504314574, 1.38061566956664894526737355602, 1.39836020346705741124725404928, 1.43558303572464063652786166770, 1.63683789689803803790375716731, 1.72493627193917410591681847926, 1.73878592638556919401688509093, 1.86949305910042269963600738626, 2.14542622047763174825906926865, 2.23109067361082671130569412224, 2.37447515356949653604174520290, 2.40391553191293171297846619487, 2.49166400752802777459498586444, 2.49344292771307493841326066102, 2.51188338649749123929196058510, 2.59674807405601397233540262213, 2.80586138128467381456020987468
Plot not available for L-functions of degree greater than 10.
