| L(s) = 1 | + 5.31e5·3-s − 2.92e8·5-s − 3.58e9·7-s + 2.82e11·9-s − 1.51e13·11-s + 1.22e12·13-s − 1.55e14·15-s + 2.51e15·17-s + 7.99e15·19-s − 1.90e15·21-s + 9.96e16·23-s − 2.12e17·25-s + 1.50e17·27-s − 2.08e18·29-s + 4.93e18·31-s − 8.03e18·33-s + 1.04e18·35-s + 1.98e19·37-s + 6.48e17·39-s + 2.24e20·41-s + 7.22e19·43-s − 8.26e19·45-s − 1.89e20·47-s − 1.32e21·49-s + 1.33e21·51-s − 2.64e21·53-s + 4.42e21·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.536·5-s − 0.0977·7-s + 1/3·9-s − 1.45·11-s + 0.0145·13-s − 0.309·15-s + 1.04·17-s + 0.828·19-s − 0.0564·21-s + 0.948·23-s − 0.712·25-s + 0.192·27-s − 1.09·29-s + 1.12·31-s − 0.838·33-s + 0.0524·35-s + 0.495·37-s + 0.00839·39-s + 1.55·41-s + 0.275·43-s − 0.178·45-s − 0.238·47-s − 0.990·49-s + 0.605·51-s − 0.739·53-s + 0.778·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p^{12} T \) |
| 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
−10.30170758799655032815189352660, −9.292804891097374034900514904691, −7.911211078669296358081844374213, −7.49998052220820775341378574056, −5.81427739622505233242516878689, −4.68585579665638431432672264691, −3.38720239314110401403519801357, −2.61978928016559819391111979494, −1.19578369101227955572375892223, 0,
1.19578369101227955572375892223, 2.61978928016559819391111979494, 3.38720239314110401403519801357, 4.68585579665638431432672264691, 5.81427739622505233242516878689, 7.49998052220820775341378574056, 7.911211078669296358081844374213, 9.292804891097374034900514904691, 10.30170758799655032815189352660
