| L(s) = 1 | − 2-s + 4-s + 2.80·5-s − 8-s − 2.80·10-s + 11-s + 16-s − 7.96·17-s − 6.48·19-s + 2.80·20-s − 22-s + 5.28·23-s + 2.87·25-s − 5.12·29-s − 8.96·31-s − 32-s + 7.96·34-s + 10.4·37-s + 6.48·38-s − 2.80·40-s + 5.09·41-s + 11.4·43-s + 44-s − 5.28·46-s − 0.322·47-s − 2.87·50-s − 8·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.25·5-s − 0.353·8-s − 0.887·10-s + 0.301·11-s + 0.250·16-s − 1.93·17-s − 1.48·19-s + 0.627·20-s − 0.213·22-s + 1.10·23-s + 0.574·25-s − 0.952·29-s − 1.61·31-s − 0.176·32-s + 1.36·34-s + 1.72·37-s + 1.05·38-s − 0.443·40-s + 0.795·41-s + 1.74·43-s + 0.150·44-s − 0.779·46-s − 0.0471·47-s − 0.406·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.80T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 0.322T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 0.871T + 59T^{2} \) |
| 61 | \( 1 - 2.15T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 - 7.44T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 1.61T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.28788622642520136789915827878, −6.60529779673015006631654424414, −6.16323737919315567129194713919, −5.50352515419386436509365950406, −4.55350280690432991381603486417, −3.85590808501459984942544076550, −2.48662868946096378495441374598, −2.23857344486020674594146839751, −1.29764244572361468263513943308, 0,
1.29764244572361468263513943308, 2.23857344486020674594146839751, 2.48662868946096378495441374598, 3.85590808501459984942544076550, 4.55350280690432991381603486417, 5.50352515419386436509365950406, 6.16323737919315567129194713919, 6.60529779673015006631654424414, 7.28788622642520136789915827878
