| L(s) = 1 | − 4.08·2-s + 8.68·4-s + 7.45·5-s + 34.0·7-s − 2.80·8-s − 30.4·10-s − 30.1·11-s − 5.30·13-s − 138.·14-s − 58.0·16-s − 63.2·17-s − 86.0·19-s + 64.7·20-s + 122.·22-s − 83.0·23-s − 69.4·25-s + 21.6·26-s + 295.·28-s + 27.8·29-s − 48.3·31-s + 259.·32-s + 258.·34-s + 253.·35-s − 358.·37-s + 351.·38-s − 20.8·40-s + 139.·41-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.08·4-s + 0.666·5-s + 1.83·7-s − 0.123·8-s − 0.963·10-s − 0.825·11-s − 0.113·13-s − 2.65·14-s − 0.906·16-s − 0.902·17-s − 1.03·19-s + 0.723·20-s + 1.19·22-s − 0.752·23-s − 0.555·25-s + 0.163·26-s + 1.99·28-s + 0.178·29-s − 0.279·31-s + 1.43·32-s + 1.30·34-s + 1.22·35-s − 1.59·37-s + 1.50·38-s − 0.0825·40-s + 0.533·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 4.08T + 8T^{2} \) |
| 5 | \( 1 - 7.45T + 125T^{2} \) |
| 7 | \( 1 - 34.0T + 343T^{2} \) |
| 11 | \( 1 + 30.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.30T + 2.19e3T^{2} \) |
| 17 | \( 1 + 63.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 83.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 27.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 48.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 608.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 361.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 733.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 280.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 239.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 149.T + 9.12e5T^{2} \) |
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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
−9.994841998406878244326588079640, −8.977802618602611659320196089005, −8.255888463660043470429432861741, −7.73977567604736657454602598701, −6.60671314280134095457151977687, −5.30509116777287721221128782732, −4.41326524051808771235073567869, −2.19286466322148765262118060720, −1.66054914771204176326416849112, 0,
1.66054914771204176326416849112, 2.19286466322148765262118060720, 4.41326524051808771235073567869, 5.30509116777287721221128782732, 6.60671314280134095457151977687, 7.73977567604736657454602598701, 8.255888463660043470429432861741, 8.977802618602611659320196089005, 9.994841998406878244326588079640
