| L(s) = 1 | − 3.09·2-s + 1.56·4-s + 0.265·5-s + 7·7-s + 19.9·8-s − 0.820·10-s − 11·11-s + 9.99·13-s − 21.6·14-s − 74.0·16-s − 103.·17-s + 54.7·19-s + 0.415·20-s + 34.0·22-s + 26.6·23-s − 124.·25-s − 30.9·26-s + 10.9·28-s + 17.2·29-s + 202.·31-s + 69.8·32-s + 320.·34-s + 1.85·35-s + 244.·37-s − 169.·38-s + 5.28·40-s − 306.·41-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.195·4-s + 0.0237·5-s + 0.377·7-s + 0.879·8-s − 0.0259·10-s − 0.301·11-s + 0.213·13-s − 0.413·14-s − 1.15·16-s − 1.47·17-s + 0.660·19-s + 0.00464·20-s + 0.329·22-s + 0.241·23-s − 0.999·25-s − 0.233·26-s + 0.0739·28-s + 0.110·29-s + 1.17·31-s + 0.385·32-s + 1.61·34-s + 0.00897·35-s + 1.08·37-s − 0.722·38-s + 0.0208·40-s − 1.16·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 3.09T + 8T^{2} \) |
| 5 | \( 1 - 0.265T + 125T^{2} \) |
| 13 | \( 1 - 9.99T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 26.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 17.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 306.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 330.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 74.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 428.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 350.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 153.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 192.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 821.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 727.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 410.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 289.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 225.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 420.T + 9.12e5T^{2} \) |
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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
−9.567872713609154145958480619535, −8.778067674729852093865236893610, −8.073886726274740451745246153757, −7.30128848993785672052713935960, −6.26239086316145199004627109373, −4.98363029724105475305789708952, −4.12710801840638612264800249143, −2.50537166313794527321017521796, −1.29306592444508136177593595062, 0,
1.29306592444508136177593595062, 2.50537166313794527321017521796, 4.12710801840638612264800249143, 4.98363029724105475305789708952, 6.26239086316145199004627109373, 7.30128848993785672052713935960, 8.073886726274740451745246153757, 8.778067674729852093865236893610, 9.567872713609154145958480619535
