| L(s) = 1 | + 2.55·2-s + 4.53·4-s − 3.91·5-s − 1.46·7-s + 6.47·8-s − 10.0·10-s + 2.24·11-s − 5.10·13-s − 3.75·14-s + 7.47·16-s − 3.83·19-s − 17.7·20-s + 5.74·22-s − 4.33·23-s + 10.3·25-s − 13.0·26-s − 6.65·28-s − 3.02·29-s − 7.87·31-s + 6.16·32-s + 5.74·35-s + 6.29·37-s − 9.80·38-s − 25.3·40-s − 0.415·41-s − 1.06·43-s + 10.1·44-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.26·4-s − 1.75·5-s − 0.554·7-s + 2.28·8-s − 3.16·10-s + 0.677·11-s − 1.41·13-s − 1.00·14-s + 1.86·16-s − 0.880·19-s − 3.96·20-s + 1.22·22-s − 0.903·23-s + 2.06·25-s − 2.55·26-s − 1.25·28-s − 0.562·29-s − 1.41·31-s + 1.08·32-s + 0.971·35-s + 1.03·37-s − 1.59·38-s − 4.00·40-s − 0.0648·41-s − 0.162·43-s + 1.53·44-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 | \( 1 - 2.55T + 2T^{2} \) |
| 5 | \( 1 + 3.91T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 + 4.33T + 23T^{2} \) |
| 29 | \( 1 + 3.02T + 29T^{2} \) |
| 31 | \( 1 + 7.87T + 31T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 + 0.415T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 + 9.91T + 47T^{2} \) |
| 53 | \( 1 - 1.05T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 1.58T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 6.94T + 71T^{2} \) |
| 73 | \( 1 + 0.879T + 73T^{2} \) |
| 79 | \( 1 + 3.93T + 79T^{2} \) |
| 83 | \( 1 - 9.29T + 83T^{2} \) |
| 89 | \( 1 + 4.38T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.119265584750715096793615290324, −7.42774103168494645755851428664, −6.86232640104883908876609888212, −6.12328032827204667253878666148, −5.07197447496121184461912257890, −4.36652185495163190613837015169, −3.80788114071643718360131388334, −3.15418313569798818542643034155, −2.08363459972587203449619683754, 0,
2.08363459972587203449619683754, 3.15418313569798818542643034155, 3.80788114071643718360131388334, 4.36652185495163190613837015169, 5.07197447496121184461912257890, 6.12328032827204667253878666148, 6.86232640104883908876609888212, 7.42774103168494645755851428664, 8.119265584750715096793615290324
