L(s) = 1 | − 3.24·3-s − 3.27·5-s + 3.90·7-s + 7.55·9-s + 3.39·11-s + 0.485·13-s + 10.6·15-s − 17-s − 3.70·19-s − 12.6·21-s + 6.21·23-s + 5.71·25-s − 14.8·27-s + 3.88·29-s − 4.41·31-s − 11.0·33-s − 12.7·35-s − 10.0·37-s − 1.57·39-s − 1.01·41-s + 10.8·43-s − 24.7·45-s − 11.2·47-s + 8.25·49-s + 3.24·51-s − 9.21·53-s − 11.1·55-s + ⋯ |
L(s) = 1 | − 1.87·3-s − 1.46·5-s + 1.47·7-s + 2.51·9-s + 1.02·11-s + 0.134·13-s + 2.74·15-s − 0.242·17-s − 0.849·19-s − 2.76·21-s + 1.29·23-s + 1.14·25-s − 2.85·27-s + 0.720·29-s − 0.793·31-s − 1.92·33-s − 2.16·35-s − 1.65·37-s − 0.252·39-s − 0.159·41-s + 1.64·43-s − 3.68·45-s − 1.64·47-s + 1.17·49-s + 0.455·51-s − 1.26·53-s − 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 - 0.485T + 13T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 + 4.41T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 0.0650T + 67T^{2} \) |
| 71 | \( 1 - 6.83T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 0.493T + 83T^{2} \) |
| 89 | \( 1 - 4.81T + 89T^{2} \) |
| 97 | \( 1 - 4.67T + 97T^{2} \) |
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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
−7.27871752603291666921552509158, −6.83787189673250554630523282203, −6.14568495471977316870668205158, −5.19756200551824710947723244342, −4.69096671124403209351843723376, −4.27221515500802921261009552561, −3.47443482304946433918078407726, −1.75173586141286990161274596569, −1.04307449331364630874134043064, 0,
1.04307449331364630874134043064, 1.75173586141286990161274596569, 3.47443482304946433918078407726, 4.27221515500802921261009552561, 4.69096671124403209351843723376, 5.19756200551824710947723244342, 6.14568495471977316870668205158, 6.83787189673250554630523282203, 7.27871752603291666921552509158