L(s) = 1 | − 3-s − 1.68·5-s − 2.23·7-s + 9-s − 3.03·11-s − 4.55·13-s + 1.68·15-s − 2.98·17-s − 2.14·19-s + 2.23·21-s + 1.24·23-s − 2.16·25-s − 27-s − 6.48·29-s + 0.905·31-s + 3.03·33-s + 3.75·35-s + 5.19·37-s + 4.55·39-s + 0.662·41-s − 10.7·43-s − 1.68·45-s + 6.58·47-s − 2.02·49-s + 2.98·51-s − 8.56·53-s + 5.11·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.753·5-s − 0.843·7-s + 0.333·9-s − 0.914·11-s − 1.26·13-s + 0.435·15-s − 0.724·17-s − 0.493·19-s + 0.486·21-s + 0.259·23-s − 0.432·25-s − 0.192·27-s − 1.20·29-s + 0.162·31-s + 0.528·33-s + 0.635·35-s + 0.853·37-s + 0.729·39-s + 0.103·41-s − 1.63·43-s − 0.251·45-s + 0.960·47-s − 0.289·49-s + 0.418·51-s − 1.17·53-s + 0.689·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2680019752\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2680019752\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 - 0.905T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 41 | \( 1 - 0.662T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 + 0.576T + 59T^{2} \) |
| 61 | \( 1 + 9.76T + 61T^{2} \) |
| 67 | \( 1 - 4.66T + 67T^{2} \) |
| 71 | \( 1 - 2.31T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 1.55T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.222787010360180992982986167118, −7.70441608483835403074660442874, −6.95305054114164193312477176886, −6.33879730173701072322970802565, −5.39850150575523940941213685556, −4.72803287744476540150399478558, −3.92451311055458877585008007118, −2.97880269554946201719042615092, −2.05768441220548396916584711178, −0.28395693868843407814609047578,
0.28395693868843407814609047578, 2.05768441220548396916584711178, 2.97880269554946201719042615092, 3.92451311055458877585008007118, 4.72803287744476540150399478558, 5.39850150575523940941213685556, 6.33879730173701072322970802565, 6.95305054114164193312477176886, 7.70441608483835403074660442874, 8.222787010360180992982986167118