| L(s) = 1 | − 0.688·3-s + 0.688·5-s + 7-s − 2.52·9-s + 0.903·11-s + 0.622·13-s − 0.474·15-s + 1.31·17-s − 7.05·19-s − 0.688·21-s − 23-s − 4.52·25-s + 3.80·27-s + 1.52·29-s + 3.54·31-s − 0.622·33-s + 0.688·35-s + 2.42·37-s − 0.428·39-s − 4.75·41-s − 6.23·43-s − 1.73·45-s − 9.73·47-s + 49-s − 0.903·51-s − 6.85·53-s + 0.622·55-s + ⋯ |
| L(s) = 1 | − 0.397·3-s + 0.308·5-s + 0.377·7-s − 0.841·9-s + 0.272·11-s + 0.172·13-s − 0.122·15-s + 0.317·17-s − 1.61·19-s − 0.150·21-s − 0.208·23-s − 0.905·25-s + 0.732·27-s + 0.283·29-s + 0.636·31-s − 0.108·33-s + 0.116·35-s + 0.399·37-s − 0.0686·39-s − 0.742·41-s − 0.950·43-s − 0.259·45-s − 1.42·47-s + 0.142·49-s − 0.126·51-s − 0.941·53-s + 0.0838·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.688T + 3T^{2} \) |
| 5 | \( 1 - 0.688T + 5T^{2} \) |
| 11 | \( 1 - 0.903T + 11T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 + 9.97T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 - 5.76T + 67T^{2} \) |
| 71 | \( 1 - 7.61T + 71T^{2} \) |
| 73 | \( 1 + 1.86T + 73T^{2} \) |
| 79 | \( 1 + 7.65T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 + 2.49T + 97T^{2} \) |
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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
−8.293911841101800958062576442390, −8.072595971914267518220790671114, −6.66244608941361463488974207966, −6.28169243891246596511527525802, −5.41067534238377556885450946227, −4.64903813913361715846062162401, −3.67748125144804518355314678182, −2.57689548851571416924489779325, −1.55666899099047255247100046774, 0,
1.55666899099047255247100046774, 2.57689548851571416924489779325, 3.67748125144804518355314678182, 4.64903813913361715846062162401, 5.41067534238377556885450946227, 6.28169243891246596511527525802, 6.66244608941361463488974207966, 8.072595971914267518220790671114, 8.293911841101800958062576442390
