| L(s) = 1 | + 2·2-s + 4·4-s − 12·5-s + 7·7-s + 8·8-s − 24·10-s + 22·11-s − 13·13-s + 14·14-s + 16·16-s + 2·17-s − 88·19-s − 48·20-s + 44·22-s + 80·23-s + 19·25-s − 26·26-s + 28·28-s + 22·29-s − 92·31-s + 32·32-s + 4·34-s − 84·35-s + 118·37-s − 176·38-s − 96·40-s − 324·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.07·5-s + 0.377·7-s + 0.353·8-s − 0.758·10-s + 0.603·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.0285·17-s − 1.06·19-s − 0.536·20-s + 0.426·22-s + 0.725·23-s + 0.151·25-s − 0.196·26-s + 0.188·28-s + 0.140·29-s − 0.533·31-s + 0.176·32-s + 0.0201·34-s − 0.405·35-s + 0.524·37-s − 0.751·38-s − 0.379·40-s − 1.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 88 T + p^{3} T^{2} \) |
| 23 | \( 1 - 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 22 T + p^{3} T^{2} \) |
| 31 | \( 1 + 92 T + p^{3} T^{2} \) |
| 37 | \( 1 - 118 T + p^{3} T^{2} \) |
| 41 | \( 1 + 324 T + p^{3} T^{2} \) |
| 43 | \( 1 - 84 T + p^{3} T^{2} \) |
| 47 | \( 1 - 134 T + p^{3} T^{2} \) |
| 53 | \( 1 - 194 T + p^{3} T^{2} \) |
| 59 | \( 1 + 210 T + p^{3} T^{2} \) |
| 61 | \( 1 + 470 T + p^{3} T^{2} \) |
| 67 | \( 1 + 292 T + p^{3} T^{2} \) |
| 71 | \( 1 - 66 T + p^{3} T^{2} \) |
| 73 | \( 1 + 506 T + p^{3} T^{2} \) |
| 79 | \( 1 + 776 T + p^{3} T^{2} \) |
| 83 | \( 1 - 778 T + p^{3} T^{2} \) |
| 89 | \( 1 - 920 T + p^{3} T^{2} \) |
| 97 | \( 1 + 490 T + p^{3} T^{2} \) |
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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
−8.491223639369613749498258377224, −7.71307624005298554202318282142, −7.01313927325803348543933439546, −6.19860942029526542382453417716, −5.13985894744309484309331862252, −4.32542510596678039775014228989, −3.72929811930419891792164549360, −2.65673066240154064080769016319, −1.42342308894314594836953390863, 0,
1.42342308894314594836953390863, 2.65673066240154064080769016319, 3.72929811930419891792164549360, 4.32542510596678039775014228989, 5.13985894744309484309331862252, 6.19860942029526542382453417716, 7.01313927325803348543933439546, 7.71307624005298554202318282142, 8.491223639369613749498258377224
