| L(s) = 1 | + 3·3-s − 6·5-s + 9·9-s + 36·11-s − 62·13-s − 18·15-s − 114·17-s + 76·19-s − 24·23-s − 89·25-s + 27·27-s + 54·29-s + 112·31-s + 108·33-s − 178·37-s − 186·39-s − 378·41-s − 172·43-s − 54·45-s + 192·47-s − 342·51-s − 402·53-s − 216·55-s + 228·57-s − 396·59-s − 254·61-s + 372·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.536·5-s + 1/3·9-s + 0.986·11-s − 1.32·13-s − 0.309·15-s − 1.62·17-s + 0.917·19-s − 0.217·23-s − 0.711·25-s + 0.192·27-s + 0.345·29-s + 0.648·31-s + 0.569·33-s − 0.790·37-s − 0.763·39-s − 1.43·41-s − 0.609·43-s − 0.178·45-s + 0.595·47-s − 0.939·51-s − 1.04·53-s − 0.529·55-s + 0.529·57-s − 0.873·59-s − 0.533·61-s + 0.709·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 24 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 192 T + p^{3} T^{2} \) |
| 53 | \( 1 + 402 T + p^{3} T^{2} \) |
| 59 | \( 1 + 396 T + p^{3} T^{2} \) |
| 61 | \( 1 + 254 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1012 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 890 T + p^{3} T^{2} \) |
| 79 | \( 1 - 80 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1638 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1010 T + p^{3} T^{2} \) |
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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
−9.675550357258769549254894310037, −9.028209042992281313336963436813, −8.086849937059166238792163391603, −7.21555397372088813299432074698, −6.45342714735365315261759535333, −4.94137639882900228606436959178, −4.11506300762397736083935286382, −2.99402942174521467071371077633, −1.75181116934142277342902817606, 0,
1.75181116934142277342902817606, 2.99402942174521467071371077633, 4.11506300762397736083935286382, 4.94137639882900228606436959178, 6.45342714735365315261759535333, 7.21555397372088813299432074698, 8.086849937059166238792163391603, 9.028209042992281313336963436813, 9.675550357258769549254894310037
