| L(s) = 1 | + 4.80·2-s + 15.0·4-s + 5.92·7-s + 33.9·8-s − 40.2·11-s + 42.1·13-s + 28.4·14-s + 42.6·16-s − 93.7·17-s − 132.·19-s − 193.·22-s − 45.1·23-s + 202.·26-s + 89.3·28-s − 68.8·29-s − 3.17·31-s − 67.1·32-s − 450.·34-s − 386.·37-s − 635.·38-s + 483.·41-s − 22.0·43-s − 606.·44-s − 217.·46-s + 96.2·47-s − 307.·49-s + 634.·52-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.88·4-s + 0.319·7-s + 1.50·8-s − 1.10·11-s + 0.898·13-s + 0.543·14-s + 0.665·16-s − 1.33·17-s − 1.59·19-s − 1.87·22-s − 0.409·23-s + 1.52·26-s + 0.602·28-s − 0.441·29-s − 0.0184·31-s − 0.370·32-s − 2.27·34-s − 1.71·37-s − 2.71·38-s + 1.84·41-s − 0.0782·43-s − 2.07·44-s − 0.695·46-s + 0.298·47-s − 0.897·49-s + 1.69·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4.80T + 8T^{2} \) |
| 7 | \( 1 - 5.92T + 343T^{2} \) |
| 11 | \( 1 + 40.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 132.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 68.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 3.17T + 2.97e4T^{2} \) |
| 37 | \( 1 + 386.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 483.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 22.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 96.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 487.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 783.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 240.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 572.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 771.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 651.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 763.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 741.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 537.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.310096767719357286489349778192, −7.34208183897784216169929789562, −6.46902796140087124306230159466, −5.92003478200018183069087358741, −5.02825656312120879338739954058, −4.35637217747431018688675339092, −3.63474148265222997588351058701, −2.53320071770666992747039982474, −1.87043891081586963457020398576, 0,
1.87043891081586963457020398576, 2.53320071770666992747039982474, 3.63474148265222997588351058701, 4.35637217747431018688675339092, 5.02825656312120879338739954058, 5.92003478200018183069087358741, 6.46902796140087124306230159466, 7.34208183897784216169929789562, 8.310096767719357286489349778192
