| L(s) = 1 | + 1.84·2-s + 1.41·4-s + 1.61·5-s + 0.152·7-s − 1.08·8-s + 2.98·10-s − 5.02·11-s − 3.94·13-s + 0.281·14-s − 4.82·16-s − 6.57·19-s + 2.28·20-s − 9.28·22-s − 3.44·23-s − 2.39·25-s − 7.28·26-s + 0.215·28-s + 2.24·29-s − 2.57·31-s − 6.75·32-s + 0.245·35-s + 10.6·37-s − 12.1·38-s − 1.74·40-s + 0.276·41-s + 6.34·43-s − 7.10·44-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.707·4-s + 0.721·5-s + 0.0575·7-s − 0.382·8-s + 0.942·10-s − 1.51·11-s − 1.09·13-s + 0.0751·14-s − 1.20·16-s − 1.50·19-s + 0.510·20-s − 1.98·22-s − 0.719·23-s − 0.479·25-s − 1.42·26-s + 0.0406·28-s + 0.416·29-s − 0.462·31-s − 1.19·32-s + 0.0415·35-s + 1.75·37-s − 1.97·38-s − 0.276·40-s + 0.0432·41-s + 0.967·43-s − 1.07·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.84T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 - 0.152T + 7T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 13 | \( 1 + 3.94T + 13T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.276T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 + 9.82T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 + 1.32T + 59T^{2} \) |
| 61 | \( 1 - 8.27T + 61T^{2} \) |
| 67 | \( 1 - 7.10T + 67T^{2} \) |
| 71 | \( 1 - 6.54T + 71T^{2} \) |
| 73 | \( 1 - 8.32T + 73T^{2} \) |
| 79 | \( 1 - 0.532T + 79T^{2} \) |
| 83 | \( 1 + 2.20T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 - 3.60T + 97T^{2} \) |
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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
−8.326443423894648933867039201535, −7.69834472879465365452179631716, −6.59216301746123140901843410752, −6.00079049419407886735336054779, −5.22568323647524601707644568159, −4.69408048878513358427455764769, −3.80094166364199026366802304800, −2.54595401424973587883987699183, −2.24511511871805715294870119558, 0,
2.24511511871805715294870119558, 2.54595401424973587883987699183, 3.80094166364199026366802304800, 4.69408048878513358427455764769, 5.22568323647524601707644568159, 6.00079049419407886735336054779, 6.59216301746123140901843410752, 7.69834472879465365452179631716, 8.326443423894648933867039201535
