| L(s) = 1 | + 1.70·2-s + 0.914·4-s − 4.06·5-s + 7-s − 1.85·8-s − 6.94·10-s + 3.27·13-s + 1.70·14-s − 4.99·16-s + 1.32·17-s + 2.16·19-s − 3.71·20-s + 1.86·23-s + 11.5·25-s + 5.59·26-s + 0.914·28-s + 0.244·29-s − 6.86·31-s − 4.81·32-s + 2.25·34-s − 4.06·35-s + 0.255·37-s + 3.69·38-s + 7.53·40-s + 5.73·41-s − 8.01·43-s + 3.18·46-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 0.457·4-s − 1.81·5-s + 0.377·7-s − 0.655·8-s − 2.19·10-s + 0.909·13-s + 0.456·14-s − 1.24·16-s + 0.320·17-s + 0.495·19-s − 0.831·20-s + 0.389·23-s + 2.30·25-s + 1.09·26-s + 0.172·28-s + 0.0453·29-s − 1.23·31-s − 0.851·32-s + 0.386·34-s − 0.687·35-s + 0.0420·37-s + 0.598·38-s + 1.19·40-s + 0.895·41-s − 1.22·43-s + 0.469·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 13 | \( 1 - 3.27T + 13T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 - 0.244T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 - 0.255T + 37T^{2} \) |
| 41 | \( 1 - 5.73T + 41T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 - 5.00T + 53T^{2} \) |
| 59 | \( 1 - 0.983T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 + 3.00T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 + 2.07T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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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
−7.35493703301229195562697276276, −6.92297985711581151653303988768, −5.89143201619683002111819465753, −5.29052607400497642887977679031, −4.49874734737930414728342420715, −3.97050976050245732993780834551, −3.44612167428702970535010807098, −2.75170159553824904383223892508, −1.23326762595996216353734153920, 0,
1.23326762595996216353734153920, 2.75170159553824904383223892508, 3.44612167428702970535010807098, 3.97050976050245732993780834551, 4.49874734737930414728342420715, 5.29052607400497642887977679031, 5.89143201619683002111819465753, 6.92297985711581151653303988768, 7.35493703301229195562697276276
