| L(s) = 1 | + 3.08·3-s + 19.0·5-s − 7·7-s − 17.4·9-s − 11·11-s − 5.08·13-s + 58.8·15-s − 19.6·17-s − 136.·19-s − 21.5·21-s − 55.6·23-s + 239.·25-s − 137.·27-s − 24.4·29-s − 209.·31-s − 33.9·33-s − 133.·35-s − 196.·37-s − 15.6·39-s + 118.·41-s + 187.·43-s − 333.·45-s − 559.·47-s + 49·49-s − 60.6·51-s + 403.·53-s − 209.·55-s + ⋯ |
| L(s) = 1 | + 0.593·3-s + 1.70·5-s − 0.377·7-s − 0.648·9-s − 0.301·11-s − 0.108·13-s + 1.01·15-s − 0.280·17-s − 1.64·19-s − 0.224·21-s − 0.504·23-s + 1.91·25-s − 0.977·27-s − 0.156·29-s − 1.21·31-s − 0.178·33-s − 0.645·35-s − 0.872·37-s − 0.0643·39-s + 0.449·41-s + 0.663·43-s − 1.10·45-s − 1.73·47-s + 0.142·49-s − 0.166·51-s + 1.04·53-s − 0.514·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 3.08T + 27T^{2} \) |
| 5 | \( 1 - 19.0T + 125T^{2} \) |
| 13 | \( 1 + 5.08T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 55.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 187.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 559.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 403.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 120.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 808.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 116.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 882.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 83.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 599.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 857.T + 9.12e5T^{2} \) |
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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
−9.001339525790430350907239561191, −8.398875429941468690821046528509, −7.23609262707844156487492144596, −6.19347569249215205595642293591, −5.80505060418187704146043104299, −4.72451780339842022060132015999, −3.40934816556231088951933922201, −2.38617249846398248789326957124, −1.82666366775102925304556587818, 0,
1.82666366775102925304556587818, 2.38617249846398248789326957124, 3.40934816556231088951933922201, 4.72451780339842022060132015999, 5.80505060418187704146043104299, 6.19347569249215205595642293591, 7.23609262707844156487492144596, 8.398875429941468690821046528509, 9.001339525790430350907239561191
