| L(s) = 1 | + 0.642·3-s − 5-s + 3.58·7-s − 2.58·9-s − 1.35·13-s − 0.642·15-s + 5.58·17-s − 19-s + 2.30·21-s − 4.87·23-s + 25-s − 3.58·27-s − 9.58·29-s − 7.17·31-s − 3.58·35-s + 0.945·37-s − 0.871·39-s + 10.4·41-s + 2.71·43-s + 2.58·45-s − 5.89·47-s + 5.87·49-s + 3.58·51-s − 9.81·53-s − 0.642·57-s − 10.1·59-s − 3.28·61-s + ⋯ |
| L(s) = 1 | + 0.370·3-s − 0.447·5-s + 1.35·7-s − 0.862·9-s − 0.376·13-s − 0.165·15-s + 1.35·17-s − 0.229·19-s + 0.502·21-s − 1.01·23-s + 0.200·25-s − 0.690·27-s − 1.78·29-s − 1.28·31-s − 0.606·35-s + 0.155·37-s − 0.139·39-s + 1.63·41-s + 0.414·43-s + 0.385·45-s − 0.859·47-s + 0.838·49-s + 0.502·51-s − 1.34·53-s − 0.0850·57-s − 1.32·59-s − 0.420·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.642T + 3T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 - 5.58T + 17T^{2} \) |
| 23 | \( 1 + 4.87T + 23T^{2} \) |
| 29 | \( 1 + 9.58T + 29T^{2} \) |
| 31 | \( 1 + 7.17T + 31T^{2} \) |
| 37 | \( 1 - 0.945T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 + 5.89T + 47T^{2} \) |
| 53 | \( 1 + 9.81T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 3.28T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 6.45T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.76594381346805826000289704132, −7.45962995331873622906375596586, −6.11369829933117776505833896818, −5.55074601115719497442076162762, −4.84967987306799960429711169877, −3.97125873834504043717718322731, −3.28742650705999088550005517886, −2.26073590758127434110668982171, −1.47421181480811057726730724861, 0,
1.47421181480811057726730724861, 2.26073590758127434110668982171, 3.28742650705999088550005517886, 3.97125873834504043717718322731, 4.84967987306799960429711169877, 5.55074601115719497442076162762, 6.11369829933117776505833896818, 7.45962995331873622906375596586, 7.76594381346805826000289704132
