| L(s) = 1 | + 4.09e3·2-s + 1.67e7·4-s + 2.92e8·5-s + 3.58e9·7-s + 6.87e10·8-s + 1.19e12·10-s − 1.51e13·11-s + 1.22e12·13-s + 1.46e13·14-s + 2.81e14·16-s − 2.51e15·17-s − 7.99e15·19-s + 4.91e15·20-s − 6.18e16·22-s + 9.96e16·23-s − 2.12e17·25-s + 5.00e15·26-s + 6.00e16·28-s + 2.08e18·29-s − 4.93e18·31-s + 1.15e18·32-s − 1.03e19·34-s + 1.04e18·35-s + 1.98e19·37-s − 3.27e19·38-s + 2.01e19·40-s − 2.24e20·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.536·5-s + 0.0977·7-s + 0.353·8-s + 0.379·10-s − 1.45·11-s + 0.0145·13-s + 0.0691·14-s + 1/4·16-s − 1.04·17-s − 0.828·19-s + 0.268·20-s − 1.02·22-s + 0.948·23-s − 0.712·25-s + 0.0102·26-s + 0.0488·28-s + 1.09·29-s − 1.12·31-s + 0.176·32-s − 0.741·34-s + 0.0524·35-s + 0.495·37-s − 0.585·38-s + 0.189·40-s − 1.55·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{12} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 11710194 p^{2} T + p^{25} T^{2} \) |
| 7 | \( 1 - 73074368 p^{2} T + p^{25} T^{2} \) |
| 11 | \( 1 + 1373779385292 p T + p^{25} T^{2} \) |
| 13 | \( 1 - 1221071681246 T + p^{25} T^{2} \) |
| 17 | \( 1 + 2518250853863682 T + p^{25} T^{2} \) |
| 19 | \( 1 + 7992693407413060 T + p^{25} T^{2} \) |
| 23 | \( 1 - 99645642629247624 T + p^{25} T^{2} \) |
| 29 | \( 1 - 2080672742244316890 T + p^{25} T^{2} \) |
| 31 | \( 1 + 4937672075835729208 T + p^{25} T^{2} \) |
| 37 | \( 1 - 19829154107621718182 T + p^{25} T^{2} \) |
| 41 | \( 1 + \)\(22\!\cdots\!42\)\( T + p^{25} T^{2} \) |
| 43 | \( 1 + 72221008334482349884 T + p^{25} T^{2} \) |
| 47 | \( 1 + \)\(18\!\cdots\!92\)\( T + p^{25} T^{2} \) |
| 53 | \( 1 - \)\(26\!\cdots\!74\)\( T + p^{25} T^{2} \) |
| 59 | \( 1 - \)\(16\!\cdots\!40\)\( T + p^{25} T^{2} \) |
| 61 | \( 1 + \)\(35\!\cdots\!38\)\( T + p^{25} T^{2} \) |
| 67 | \( 1 - \)\(10\!\cdots\!92\)\( T + p^{25} T^{2} \) |
| 71 | \( 1 + \)\(73\!\cdots\!12\)\( T + p^{25} T^{2} \) |
| 73 | \( 1 + \)\(26\!\cdots\!74\)\( T + p^{25} T^{2} \) |
| 79 | \( 1 + \)\(10\!\cdots\!40\)\( T + p^{25} T^{2} \) |
| 83 | \( 1 + \)\(15\!\cdots\!96\)\( T + p^{25} T^{2} \) |
| 89 | \( 1 + \)\(21\!\cdots\!90\)\( T + p^{25} T^{2} \) |
| 97 | \( 1 + \)\(44\!\cdots\!58\)\( T + p^{25} T^{2} \) |
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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
−12.83605385759259503556387876700, −11.22537417156832596022110802557, −10.14684891054033031972836183624, −8.450656885706566060080891961444, −6.94152731158870908790183773330, −5.62568803381087301199772431691, −4.54334726470719239072445583530, −2.89602380884521572167507375049, −1.84481845224444664544179333228, 0,
1.84481845224444664544179333228, 2.89602380884521572167507375049, 4.54334726470719239072445583530, 5.62568803381087301199772431691, 6.94152731158870908790183773330, 8.450656885706566060080891961444, 10.14684891054033031972836183624, 11.22537417156832596022110802557, 12.83605385759259503556387876700
