| L(s) = 1 | − 2.09·2-s + 3-s + 2.40·4-s + 3.15·5-s − 2.09·6-s − 7-s − 0.839·8-s + 9-s − 6.62·10-s + 2.40·12-s + 2.12·13-s + 2.09·14-s + 3.15·15-s − 3.03·16-s − 4.79·17-s − 2.09·18-s + 1.53·19-s + 7.58·20-s − 21-s + 5.11·23-s − 0.839·24-s + 4.98·25-s − 4.46·26-s + 27-s − 2.40·28-s − 0.958·29-s − 6.62·30-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 0.577·3-s + 1.20·4-s + 1.41·5-s − 0.856·6-s − 0.377·7-s − 0.296·8-s + 0.333·9-s − 2.09·10-s + 0.692·12-s + 0.590·13-s + 0.560·14-s + 0.815·15-s − 0.759·16-s − 1.16·17-s − 0.494·18-s + 0.352·19-s + 1.69·20-s − 0.218·21-s + 1.06·23-s − 0.171·24-s + 0.996·25-s − 0.875·26-s + 0.192·27-s − 0.453·28-s − 0.177·29-s − 1.21·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.378158008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378158008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 + 0.958T + 29T^{2} \) |
| 31 | \( 1 + 7.22T + 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 - 0.266T + 41T^{2} \) |
| 43 | \( 1 - 9.28T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 0.945T + 53T^{2} \) |
| 59 | \( 1 + 9.55T + 59T^{2} \) |
| 61 | \( 1 - 8.55T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 7.72T + 73T^{2} \) |
| 79 | \( 1 - 9.31T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 7.08T + 97T^{2} \) |
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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
−9.153628844723754928061120081725, −8.470500248241099182910794524018, −7.48292113573765191623002903726, −6.83829429852530874487274951159, −6.10576943929449982797032140895, −5.14887664369057031026979771661, −3.91619716128855402548077450092, −2.61196705185728362051280744121, −1.98361920140482984302870388501, −0.934070198580870349618378020094,
0.934070198580870349618378020094, 1.98361920140482984302870388501, 2.61196705185728362051280744121, 3.91619716128855402548077450092, 5.14887664369057031026979771661, 6.10576943929449982797032140895, 6.83829429852530874487274951159, 7.48292113573765191623002903726, 8.470500248241099182910794524018, 9.153628844723754928061120081725
