L(s) = 1 | + 0.246·2-s − 1.93·4-s + 1.24·7-s − 0.972·8-s − 2.56·11-s + 4.68·13-s + 0.307·14-s + 3.63·16-s + 5.83·17-s + 4.16·19-s − 0.634·22-s + 1.60·23-s + 1.15·26-s − 2.41·28-s + 3.21·29-s − 9.19·31-s + 2.84·32-s + 1.44·34-s + 1.27·37-s + 1.02·38-s − 8.69·41-s + 3.88·43-s + 4.98·44-s + 0.396·46-s − 3.20·47-s − 5.44·49-s − 9.07·52-s + ⋯ |
L(s) = 1 | + 0.174·2-s − 0.969·4-s + 0.471·7-s − 0.343·8-s − 0.774·11-s + 1.29·13-s + 0.0822·14-s + 0.909·16-s + 1.41·17-s + 0.956·19-s − 0.135·22-s + 0.334·23-s + 0.226·26-s − 0.456·28-s + 0.597·29-s − 1.65·31-s + 0.502·32-s + 0.247·34-s + 0.210·37-s + 0.166·38-s − 1.35·41-s + 0.591·43-s + 0.751·44-s + 0.0584·46-s − 0.468·47-s − 0.777·49-s − 1.25·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.855488824\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.855488824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 0.246T + 2T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 - 1.27T + 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 - 3.88T + 43T^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 6.39T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 2.83T + 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 + 0.0305T + 97T^{2} \) |
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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
−8.064221970567513712528365137476, −7.73329027860680182017997642371, −6.66854518033288118632320232618, −5.65388872085827863203309743492, −5.32600577973295216213921394595, −4.59090757082287673532074520190, −3.50853527140452202814292572056, −3.22557522456088259888086503157, −1.68361734046066454933135768735, −0.75532988907201590714008960982,
0.75532988907201590714008960982, 1.68361734046066454933135768735, 3.22557522456088259888086503157, 3.50853527140452202814292572056, 4.59090757082287673532074520190, 5.32600577973295216213921394595, 5.65388872085827863203309743492, 6.66854518033288118632320232618, 7.73329027860680182017997642371, 8.064221970567513712528365137476