| L(s) = 1 | + 0.854·2-s − 1.26·4-s − 3.95·5-s − 4.38·7-s − 2.79·8-s − 3.38·10-s + 2.89·11-s − 2.95·13-s − 3.74·14-s + 0.149·16-s + 3.86·17-s + 2.14·19-s + 5.02·20-s + 2.47·22-s + 0.0986·23-s + 10.6·25-s − 2.52·26-s + 5.56·28-s + 0.0777·31-s + 5.71·32-s + 3.30·34-s + 17.3·35-s + 10.4·37-s + 1.83·38-s + 11.0·40-s − 6.48·41-s − 1.79·43-s + ⋯ |
| L(s) = 1 | + 0.604·2-s − 0.634·4-s − 1.77·5-s − 1.65·7-s − 0.988·8-s − 1.07·10-s + 0.872·11-s − 0.818·13-s − 1.00·14-s + 0.0373·16-s + 0.936·17-s + 0.492·19-s + 1.12·20-s + 0.527·22-s + 0.0205·23-s + 2.13·25-s − 0.494·26-s + 1.05·28-s + 0.0139·31-s + 1.01·32-s + 0.565·34-s + 2.93·35-s + 1.71·37-s + 0.297·38-s + 1.74·40-s − 1.01·41-s − 0.273·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 0.854T + 2T^{2} \) |
| 5 | \( 1 + 3.95T + 5T^{2} \) |
| 7 | \( 1 + 4.38T + 7T^{2} \) |
| 11 | \( 1 - 2.89T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 3.86T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 23 | \( 1 - 0.0986T + 23T^{2} \) |
| 31 | \( 1 - 0.0777T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 + 9.87T + 47T^{2} \) |
| 53 | \( 1 + 7.41T + 53T^{2} \) |
| 59 | \( 1 + 2.09T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 - 4.65T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 6.78T + 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.61847449250680546439367642330, −6.66021674543441020955644797650, −6.31274724351609267844491935090, −5.19497424914620288571915854072, −4.60727315292324814480393156481, −3.65857316954239732486641485917, −3.53828243152346200372266050557, −2.80402220373610882413411301038, −0.845043818549229769684692642391, 0,
0.845043818549229769684692642391, 2.80402220373610882413411301038, 3.53828243152346200372266050557, 3.65857316954239732486641485917, 4.60727315292324814480393156481, 5.19497424914620288571915854072, 6.31274724351609267844491935090, 6.66021674543441020955644797650, 7.61847449250680546439367642330
