| L(s) = 1 | − 0.692·3-s + 3.18·5-s + 3.79·7-s − 2.52·9-s − 2.76·11-s + 2.89·13-s − 2.20·15-s − 7.05·17-s − 5.44·19-s − 2.62·21-s − 5.85·23-s + 5.16·25-s + 3.82·27-s − 0.0264·29-s − 1.32·31-s + 1.91·33-s + 12.0·35-s + 9.73·37-s − 2.00·39-s + 1.29·41-s + 8.70·43-s − 8.03·45-s − 8.73·47-s + 7.40·49-s + 4.88·51-s − 6.78·53-s − 8.80·55-s + ⋯ |
| L(s) = 1 | − 0.399·3-s + 1.42·5-s + 1.43·7-s − 0.840·9-s − 0.832·11-s + 0.803·13-s − 0.569·15-s − 1.71·17-s − 1.24·19-s − 0.573·21-s − 1.22·23-s + 1.03·25-s + 0.735·27-s − 0.00491·29-s − 0.238·31-s + 0.332·33-s + 2.04·35-s + 1.59·37-s − 0.321·39-s + 0.201·41-s + 1.32·43-s − 1.19·45-s − 1.27·47-s + 1.05·49-s + 0.683·51-s − 0.932·53-s − 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 0.692T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 - 3.79T + 7T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 + 7.05T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 + 0.0264T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 - 9.73T + 37T^{2} \) |
| 41 | \( 1 - 1.29T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 + 6.78T + 53T^{2} \) |
| 59 | \( 1 + 2.43T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 8.31T + 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 + 1.12T + 73T^{2} \) |
| 79 | \( 1 + 1.98T + 79T^{2} \) |
| 83 | \( 1 + 7.57T + 83T^{2} \) |
| 89 | \( 1 + 3.29T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−7.64627328887221701434619681395, −6.32621033476748449423608544903, −6.19446708534271153497463534971, −5.50697538510099640548928808101, −4.71486619511402489628977272731, −4.24197820263054276283315665401, −2.73292552547937308629104895299, −2.16071744132985188546517813688, −1.49492753400755326606767053656, 0,
1.49492753400755326606767053656, 2.16071744132985188546517813688, 2.73292552547937308629104895299, 4.24197820263054276283315665401, 4.71486619511402489628977272731, 5.50697538510099640548928808101, 6.19446708534271153497463534971, 6.32621033476748449423608544903, 7.64627328887221701434619681395
