| L(s) = 1 | + 12·3-s − 4.33e3·5-s + 1.39e5·7-s − 1.59e6·9-s + 6.48e6·11-s − 2.25e7·13-s − 5.19e4·15-s − 2.37e7·17-s − 3.25e8·19-s + 1.67e6·21-s − 9.21e8·23-s − 1.20e9·25-s − 3.82e7·27-s − 3.86e9·29-s + 2.25e9·31-s + 7.78e7·33-s − 6.06e8·35-s + 1.82e10·37-s − 2.71e8·39-s + 3.44e10·41-s + 1.71e10·43-s + 6.90e9·45-s + 6.73e10·47-s − 7.72e10·49-s − 2.84e8·51-s − 8.72e10·53-s − 2.80e10·55-s + ⋯ |
| L(s) = 1 | + 0.00950·3-s − 0.123·5-s + 0.449·7-s − 0.999·9-s + 1.10·11-s − 1.29·13-s − 0.00117·15-s − 0.238·17-s − 1.58·19-s + 0.00427·21-s − 1.29·23-s − 0.984·25-s − 0.0190·27-s − 1.20·29-s + 0.456·31-s + 0.0104·33-s − 0.0557·35-s + 1.16·37-s − 0.0123·39-s + 1.13·41-s + 0.414·43-s + 0.123·45-s + 0.911·47-s − 0.797·49-s − 0.00226·51-s − 0.540·53-s − 0.136·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 p T + p^{13} T^{2} \) |
| 5 | \( 1 + 866 p T + p^{13} T^{2} \) |
| 7 | \( 1 - 139992 T + p^{13} T^{2} \) |
| 11 | \( 1 - 589484 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 22588034 T + p^{13} T^{2} \) |
| 17 | \( 1 + 23732270 T + p^{13} T^{2} \) |
| 19 | \( 1 + 325344836 T + p^{13} T^{2} \) |
| 23 | \( 1 + 921600632 T + p^{13} T^{2} \) |
| 29 | \( 1 + 3865879218 T + p^{13} T^{2} \) |
| 31 | \( 1 - 2253401440 T + p^{13} T^{2} \) |
| 37 | \( 1 - 18250384566 T + p^{13} T^{2} \) |
| 41 | \( 1 - 34422845322 T + p^{13} T^{2} \) |
| 43 | \( 1 - 17192501444 T + p^{13} T^{2} \) |
| 47 | \( 1 - 67371749904 T + p^{13} T^{2} \) |
| 53 | \( 1 + 1646815442 p T + p^{13} T^{2} \) |
| 59 | \( 1 + 540214518668 T + p^{13} T^{2} \) |
| 61 | \( 1 + 51276568850 T + p^{13} T^{2} \) |
| 67 | \( 1 + 25519930676 T + p^{13} T^{2} \) |
| 71 | \( 1 - 1387500699032 T + p^{13} T^{2} \) |
| 73 | \( 1 + 819049441238 T + p^{13} T^{2} \) |
| 79 | \( 1 - 4030935615344 T + p^{13} T^{2} \) |
| 83 | \( 1 + 4180823831428 T + p^{13} T^{2} \) |
| 89 | \( 1 - 2677027798266 T + p^{13} T^{2} \) |
| 97 | \( 1 + 14039464316446 T + p^{13} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.99439117238226626602963460791, −14.13218259630179337233437949187, −12.30298589159561679674723887125, −11.17094040229553414726350288804, −9.393786039688065006566494636838, −7.916655757161315094635282443481, −6.09427755367712144828389261406, −4.24167625767759190340759521571, −2.18131556477790340318315289755, 0,
2.18131556477790340318315289755, 4.24167625767759190340759521571, 6.09427755367712144828389261406, 7.916655757161315094635282443481, 9.393786039688065006566494636838, 11.17094040229553414726350288804, 12.30298589159561679674723887125, 14.13218259630179337233437949187, 14.99439117238226626602963460791
