| L(s) = 1 | + 1.82·2-s + 1.31·4-s + 5-s − 0.729·7-s − 1.24·8-s + 1.82·10-s − 0.729·11-s + 3.38·13-s − 1.32·14-s − 4.90·16-s + 5.74·17-s + 6.11·19-s + 1.31·20-s − 1.32·22-s + 9.48·23-s + 25-s + 6.15·26-s − 0.957·28-s + 29-s + 5.48·31-s − 6.42·32-s + 10.4·34-s − 0.729·35-s − 10.2·37-s + 11.1·38-s − 1.24·40-s + 11.3·41-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.656·4-s + 0.447·5-s − 0.275·7-s − 0.441·8-s + 0.575·10-s − 0.219·11-s + 0.938·13-s − 0.354·14-s − 1.22·16-s + 1.39·17-s + 1.40·19-s + 0.293·20-s − 0.282·22-s + 1.97·23-s + 0.200·25-s + 1.20·26-s − 0.180·28-s + 0.185·29-s + 0.985·31-s − 1.13·32-s + 1.79·34-s − 0.123·35-s − 1.69·37-s + 1.80·38-s − 0.197·40-s + 1.76·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.449252771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.449252771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 7 | \( 1 + 0.729T + 7T^{2} \) |
| 11 | \( 1 + 0.729T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 - 6.11T + 19T^{2} \) |
| 23 | \( 1 - 9.48T + 23T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.89T + 47T^{2} \) |
| 53 | \( 1 + 8.14T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 - 4.83T + 71T^{2} \) |
| 73 | \( 1 - 6.29T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 0.0848T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
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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
−9.595574847194210332318906549945, −8.989283979204674971476991372227, −7.892762914151509364177961979057, −6.88022676623477708125217688283, −6.07866951918344313202125584355, −5.34064961383050596117241062739, −4.71050638380219598717202588461, −3.29358622387056447369279713306, −3.08369922037493253462869783834, −1.26988037798009943994208634503,
1.26988037798009943994208634503, 3.08369922037493253462869783834, 3.29358622387056447369279713306, 4.71050638380219598717202588461, 5.34064961383050596117241062739, 6.07866951918344313202125584355, 6.88022676623477708125217688283, 7.892762914151509364177961979057, 8.989283979204674971476991372227, 9.595574847194210332318906549945
