| L(s) = 1 | + 0.130·3-s − 3.71·5-s − 7-s − 2.98·9-s − 1.76·11-s − 0.485·15-s + 0.770·17-s − 3.10·19-s − 0.130·21-s + 4.59·23-s + 8.82·25-s − 0.781·27-s + 7.93·29-s + 6.03·31-s − 0.230·33-s + 3.71·35-s + 0.537·37-s − 6.33·41-s + 5.01·43-s + 11.0·45-s + 0.525·47-s + 49-s + 0.100·51-s − 1.28·53-s + 6.55·55-s − 0.406·57-s − 10.6·59-s + ⋯ |
| L(s) = 1 | + 0.0754·3-s − 1.66·5-s − 0.377·7-s − 0.994·9-s − 0.531·11-s − 0.125·15-s + 0.186·17-s − 0.713·19-s − 0.0285·21-s + 0.958·23-s + 1.76·25-s − 0.150·27-s + 1.47·29-s + 1.08·31-s − 0.0401·33-s + 0.628·35-s + 0.0884·37-s − 0.989·41-s + 0.764·43-s + 1.65·45-s + 0.0766·47-s + 0.142·49-s + 0.0141·51-s − 0.176·53-s + 0.883·55-s − 0.0538·57-s − 1.38·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.130T + 3T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 17 | \( 1 - 0.770T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 - 4.59T + 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 - 0.537T + 37T^{2} \) |
| 41 | \( 1 + 6.33T + 41T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 - 0.525T + 47T^{2} \) |
| 53 | \( 1 + 1.28T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 6.69T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 9.67T + 79T^{2} \) |
| 83 | \( 1 - 3.22T + 83T^{2} \) |
| 89 | \( 1 + 0.391T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.65703135925751725435633904796, −6.60200209226279445610943894868, −6.22517897300818669267135099561, −5.01633333148825703810669395123, −4.66831021064537850479923091677, −3.68589072495204937825123699845, −3.11576316758797465629131465119, −2.49242332057538768301305113148, −0.890314593880467838626743169799, 0,
0.890314593880467838626743169799, 2.49242332057538768301305113148, 3.11576316758797465629131465119, 3.68589072495204937825123699845, 4.66831021064537850479923091677, 5.01633333148825703810669395123, 6.22517897300818669267135099561, 6.60200209226279445610943894868, 7.65703135925751725435633904796
