| L(s) = 1 | + 34·2-s − 792·3-s − 892·4-s + 3.12e3·5-s − 2.69e4·6-s − 1.75e4·7-s − 9.99e4·8-s + 4.50e5·9-s + 1.06e5·10-s − 4.68e5·11-s + 7.06e5·12-s − 3.74e5·13-s − 5.96e5·14-s − 2.47e6·15-s − 1.57e6·16-s − 3.72e6·17-s + 1.53e7·18-s − 3.79e5·19-s − 2.78e6·20-s + 1.39e7·21-s − 1.59e7·22-s − 3.24e7·23-s + 7.91e7·24-s + 9.76e6·25-s − 1.27e7·26-s − 2.16e8·27-s + 1.56e7·28-s + ⋯ |
| L(s) = 1 | + 0.751·2-s − 1.88·3-s − 0.435·4-s + 0.447·5-s − 1.41·6-s − 0.394·7-s − 1.07·8-s + 2.54·9-s + 0.335·10-s − 0.877·11-s + 0.819·12-s − 0.279·13-s − 0.296·14-s − 0.841·15-s − 0.374·16-s − 0.636·17-s + 1.90·18-s − 0.0351·19-s − 0.194·20-s + 0.742·21-s − 0.659·22-s − 1.05·23-s + 2.02·24-s + 1/5·25-s − 0.209·26-s − 2.89·27-s + 0.171·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{5} T \) |
| good | 2 | \( 1 - 17 p T + p^{11} T^{2} \) |
| 3 | \( 1 + 88 p^{2} T + p^{11} T^{2} \) |
| 7 | \( 1 + 2508 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 468788 T + p^{11} T^{2} \) |
| 13 | \( 1 + 374042 T + p^{11} T^{2} \) |
| 17 | \( 1 + 3724286 T + p^{11} T^{2} \) |
| 19 | \( 1 + 379460 T + p^{11} T^{2} \) |
| 23 | \( 1 + 32458092 T + p^{11} T^{2} \) |
| 29 | \( 1 - 69696710 T + p^{11} T^{2} \) |
| 31 | \( 1 - 171448632 T + p^{11} T^{2} \) |
| 37 | \( 1 + 291340546 T + p^{11} T^{2} \) |
| 41 | \( 1 - 191343242 T + p^{11} T^{2} \) |
| 43 | \( 1 + 1759857392 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1623469924 T + p^{11} T^{2} \) |
| 53 | \( 1 + 644888642 T + p^{11} T^{2} \) |
| 59 | \( 1 - 925569220 T + p^{11} T^{2} \) |
| 61 | \( 1 + 10898589338 T + p^{11} T^{2} \) |
| 67 | \( 1 - 3795674064 T + p^{11} T^{2} \) |
| 71 | \( 1 + 22966943728 T + p^{11} T^{2} \) |
| 73 | \( 1 - 9880820458 T + p^{11} T^{2} \) |
| 79 | \( 1 + 20768886240 T + p^{11} T^{2} \) |
| 83 | \( 1 - 3204862008 T + p^{11} T^{2} \) |
| 89 | \( 1 - 63176321130 T + p^{11} T^{2} \) |
| 97 | \( 1 - 126494473874 T + p^{11} T^{2} \) |
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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
−21.39545644178479372733891133718, −18.39713450807012559861519135838, −17.40012894882809944623073643587, −15.76745574902438742059475479669, −13.31948521423817325374611159168, −12.07482892919987191455384813130, −10.18108061715913802952885894831, −6.20981079042288953304728619848, −4.83549500016447341291994353606, 0,
4.83549500016447341291994353606, 6.20981079042288953304728619848, 10.18108061715913802952885894831, 12.07482892919987191455384813130, 13.31948521423817325374611159168, 15.76745574902438742059475479669, 17.40012894882809944623073643587, 18.39713450807012559861519135838, 21.39545644178479372733891133718
