| L(s) = 1 | + 3-s + 1.73·7-s + 9-s − 3.98·11-s − 1.30·13-s − 0.895·17-s − 5.46·19-s + 1.73·21-s + 1.06·23-s + 27-s + 3.43·29-s + 31-s − 3.98·33-s − 6.25·37-s − 1.30·39-s + 4.68·41-s + 12.1·43-s − 1.61·47-s − 3.99·49-s − 0.895·51-s + 10.5·53-s − 5.46·57-s − 9.43·59-s − 0.198·61-s + 1.73·63-s + 1.79·67-s + 1.06·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.655·7-s + 0.333·9-s − 1.20·11-s − 0.361·13-s − 0.217·17-s − 1.25·19-s + 0.378·21-s + 0.221·23-s + 0.192·27-s + 0.637·29-s + 0.179·31-s − 0.693·33-s − 1.02·37-s − 0.208·39-s + 0.731·41-s + 1.84·43-s − 0.235·47-s − 0.570·49-s − 0.125·51-s + 1.44·53-s − 0.723·57-s − 1.22·59-s − 0.0254·61-s + 0.218·63-s + 0.219·67-s + 0.127·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 0.895T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 37 | \( 1 + 6.25T + 37T^{2} \) |
| 41 | \( 1 - 4.68T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 9.43T + 59T^{2} \) |
| 61 | \( 1 + 0.198T + 61T^{2} \) |
| 67 | \( 1 - 1.79T + 67T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 - 1.54T + 83T^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−7.55111089489635604970920730416, −6.81898888051340858850318159116, −5.98265893940007347740981197446, −5.19488992701613230938046100586, −4.55918787937564680235464907387, −3.92786558450794443583176487519, −2.76539490946781278674137271235, −2.39385256259299952379976075619, −1.37411723342810102381454483626, 0,
1.37411723342810102381454483626, 2.39385256259299952379976075619, 2.76539490946781278674137271235, 3.92786558450794443583176487519, 4.55918787937564680235464907387, 5.19488992701613230938046100586, 5.98265893940007347740981197446, 6.81898888051340858850318159116, 7.55111089489635604970920730416
