| L(s) = 1 | − 1.49·3-s − 4.94·7-s − 0.758·9-s + 1.29·11-s + 0.820·13-s − 2.18·17-s + 6.43·19-s + 7.39·21-s + 2.21·23-s + 5.62·27-s − 5.33·29-s + 0.708·31-s − 1.93·33-s − 7.15·37-s − 1.22·39-s + 10.1·41-s + 5.26·43-s + 8.04·47-s + 17.4·49-s + 3.27·51-s − 13.2·53-s − 9.63·57-s + 9.59·59-s − 6.54·61-s + 3.74·63-s − 7.32·67-s − 3.31·69-s + ⋯ |
| L(s) = 1 | − 0.864·3-s − 1.86·7-s − 0.252·9-s + 0.390·11-s + 0.227·13-s − 0.531·17-s + 1.47·19-s + 1.61·21-s + 0.461·23-s + 1.08·27-s − 0.990·29-s + 0.127·31-s − 0.337·33-s − 1.17·37-s − 0.196·39-s + 1.58·41-s + 0.803·43-s + 1.17·47-s + 2.48·49-s + 0.459·51-s − 1.81·53-s − 1.27·57-s + 1.24·59-s − 0.837·61-s + 0.472·63-s − 0.894·67-s − 0.399·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 7 | \( 1 + 4.94T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 - 0.820T + 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 19 | \( 1 - 6.43T + 19T^{2} \) |
| 23 | \( 1 - 2.21T + 23T^{2} \) |
| 29 | \( 1 + 5.33T + 29T^{2} \) |
| 31 | \( 1 - 0.708T + 31T^{2} \) |
| 37 | \( 1 + 7.15T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 8.04T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 9.59T + 59T^{2} \) |
| 61 | \( 1 + 6.54T + 61T^{2} \) |
| 67 | \( 1 + 7.32T + 67T^{2} \) |
| 71 | \( 1 + 0.640T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−7.65970981440791600372662843028, −6.97944037431830935244820779751, −6.36267341026989855640062092660, −5.80974639453848002896701814503, −5.19058994554824910249048189565, −4.05082521886200162424518995909, −3.31821156458309520742639563210, −2.59000174468462657936370179719, −1.01588070465399631486102060000, 0,
1.01588070465399631486102060000, 2.59000174468462657936370179719, 3.31821156458309520742639563210, 4.05082521886200162424518995909, 5.19058994554824910249048189565, 5.80974639453848002896701814503, 6.36267341026989855640062092660, 6.97944037431830935244820779751, 7.65970981440791600372662843028
