| L(s) = 1 | − 0.538·3-s − 0.810·5-s − 2.70·9-s + 4.92·11-s − 0.0654·13-s + 0.436·15-s + 1.24·17-s − 3.82·19-s − 1.71·23-s − 4.34·25-s + 3.07·27-s + 3.89·29-s − 3.19·31-s − 2.65·33-s − 0.0930·37-s + 0.0352·39-s + 41-s + 10.5·43-s + 2.19·45-s + 0.959·47-s − 0.668·51-s − 2.72·53-s − 3.99·55-s + 2.06·57-s − 11.1·59-s + 10.8·61-s + 0.0530·65-s + ⋯ |
| L(s) = 1 | − 0.311·3-s − 0.362·5-s − 0.903·9-s + 1.48·11-s − 0.0181·13-s + 0.112·15-s + 0.300·17-s − 0.877·19-s − 0.358·23-s − 0.868·25-s + 0.592·27-s + 0.724·29-s − 0.574·31-s − 0.462·33-s − 0.0152·37-s + 0.00564·39-s + 0.156·41-s + 1.61·43-s + 0.327·45-s + 0.139·47-s − 0.0935·51-s − 0.374·53-s − 0.538·55-s + 0.272·57-s − 1.45·59-s + 1.38·61-s + 0.00657·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 0.538T + 3T^{2} \) |
| 5 | \( 1 + 0.810T + 5T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 + 0.0654T + 13T^{2} \) |
| 17 | \( 1 - 1.24T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 0.0930T + 37T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 0.959T + 47T^{2} \) |
| 53 | \( 1 + 2.72T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 5.60T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 9.99T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−7.43045044025938203277352148945, −6.74146165781342324970462344840, −6.02594439126194190444464188001, −5.62690096779509881842957372370, −4.48281844146463893900904956610, −4.01635526467165575057212745886, −3.17915171333207557866953770246, −2.22003233996368269913348794272, −1.16356464047781308304607556354, 0,
1.16356464047781308304607556354, 2.22003233996368269913348794272, 3.17915171333207557866953770246, 4.01635526467165575057212745886, 4.48281844146463893900904956610, 5.62690096779509881842957372370, 6.02594439126194190444464188001, 6.74146165781342324970462344840, 7.43045044025938203277352148945
