| L(s) = 1 | − 3-s − 1.61·7-s + 9-s + 1.61·11-s − 13-s − 6.17·17-s + 5.17·19-s + 1.61·21-s − 8.34·23-s − 27-s + 29-s + 1.61·31-s − 1.61·33-s + 1.17·37-s + 39-s − 3.17·41-s + 2·43-s + 1.55·47-s − 4.39·49-s + 6.17·51-s + 7.34·53-s − 5.17·57-s − 4.72·59-s + 10.5·61-s − 1.61·63-s − 4.78·67-s + 8.34·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.609·7-s + 0.333·9-s + 0.486·11-s − 0.277·13-s − 1.49·17-s + 1.18·19-s + 0.352·21-s − 1.73·23-s − 0.192·27-s + 0.185·29-s + 0.289·31-s − 0.280·33-s + 0.192·37-s + 0.160·39-s − 0.495·41-s + 0.304·43-s + 0.227·47-s − 0.628·49-s + 0.864·51-s + 1.00·53-s − 0.684·57-s − 0.615·59-s + 1.34·61-s − 0.203·63-s − 0.584·67-s + 1.00·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.064094386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.064094386\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 1.61T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 1.55T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + 4.72T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 4.78T + 67T^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 - 0.773T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 - 8.72T + 83T^{2} \) |
| 89 | \( 1 + 6.34T + 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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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
−7.74251962885662933253566089443, −7.03710260778236655721689099511, −6.42926703324054260830791983073, −5.90468482326890288913211060871, −5.07015838566720397000186970651, −4.29993548643585845828494476035, −3.66033200964660701696006502701, −2.64454088268343094660779094739, −1.74600218016868689569647449956, −0.52094666811292158527118134641,
0.52094666811292158527118134641, 1.74600218016868689569647449956, 2.64454088268343094660779094739, 3.66033200964660701696006502701, 4.29993548643585845828494476035, 5.07015838566720397000186970651, 5.90468482326890288913211060871, 6.42926703324054260830791983073, 7.03710260778236655721689099511, 7.74251962885662933253566089443
