| L(s) = 1 | + 1.40·2-s − 0.0305·4-s + 0.939·5-s + 3.20·7-s − 2.84·8-s + 1.31·10-s + 2.93·11-s − 0.838·13-s + 4.50·14-s − 3.93·16-s + 1.31·17-s + 1.38·19-s − 0.0287·20-s + 4.11·22-s + 5.00·23-s − 4.11·25-s − 1.17·26-s − 0.0981·28-s − 4.89·29-s + 10.1·31-s + 0.173·32-s + 1.84·34-s + 3.01·35-s + 5.39·37-s + 1.94·38-s − 2.67·40-s − 7.45·41-s + ⋯ |
| L(s) = 1 | + 0.992·2-s − 0.0152·4-s + 0.420·5-s + 1.21·7-s − 1.00·8-s + 0.416·10-s + 0.884·11-s − 0.232·13-s + 1.20·14-s − 0.984·16-s + 0.318·17-s + 0.317·19-s − 0.00642·20-s + 0.878·22-s + 1.04·23-s − 0.823·25-s − 0.230·26-s − 0.0185·28-s − 0.909·29-s + 1.82·31-s + 0.0305·32-s + 0.315·34-s + 0.509·35-s + 0.886·37-s + 0.315·38-s − 0.423·40-s − 1.16·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.378660285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.378660285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 1.40T + 2T^{2} \) |
| 5 | \( 1 - 0.939T + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 0.838T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 5.39T + 37T^{2} \) |
| 41 | \( 1 + 7.45T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 7.10T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 8.90T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 - 5.27T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 3.41T + 97T^{2} \) |
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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
−9.224848336487931099205427327750, −8.258783724387270228093892051342, −7.51950936732832876534779342319, −6.43294616224029386113658787813, −5.77492509467510682464798596939, −4.93148839133695962862169590713, −4.40225062736463988894585182923, −3.45087183897882165516843250927, −2.36572918391735230738861633985, −1.14289822221131708816077949759,
1.14289822221131708816077949759, 2.36572918391735230738861633985, 3.45087183897882165516843250927, 4.40225062736463988894585182923, 4.93148839133695962862169590713, 5.77492509467510682464798596939, 6.43294616224029386113658787813, 7.51950936732832876534779342319, 8.258783724387270228093892051342, 9.224848336487931099205427327750
