L(s) = 1 | + 2.65·2-s + 5.03·4-s − 3.62·5-s − 2.98·7-s + 8.05·8-s − 9.60·10-s + 1.77·11-s + 5.96·13-s − 7.91·14-s + 11.2·16-s + 5.47·17-s + 5.48·19-s − 18.2·20-s + 4.72·22-s − 2.11·23-s + 8.11·25-s + 15.8·26-s − 15.0·28-s − 2.88·29-s + 2.48·31-s + 13.8·32-s + 14.5·34-s + 10.8·35-s + 5.90·37-s + 14.5·38-s − 29.1·40-s + 2.24·41-s + ⋯ |
L(s) = 1 | + 1.87·2-s + 2.51·4-s − 1.61·5-s − 1.12·7-s + 2.84·8-s − 3.03·10-s + 0.536·11-s + 1.65·13-s − 2.11·14-s + 2.82·16-s + 1.32·17-s + 1.25·19-s − 4.07·20-s + 1.00·22-s − 0.441·23-s + 1.62·25-s + 3.10·26-s − 2.83·28-s − 0.535·29-s + 0.445·31-s + 2.44·32-s + 2.48·34-s + 1.82·35-s + 0.970·37-s + 2.36·38-s − 4.61·40-s + 0.349·41-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
\(L(1)\) |
\(\approx\) |
\(4.637061732\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.637061732\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.65T + 2T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 - 5.96T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 - 5.48T + 19T^{2} \) |
| 23 | \( 1 + 2.11T + 23T^{2} \) |
| 29 | \( 1 + 2.88T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 - 0.954T + 61T^{2} \) |
| 67 | \( 1 + 4.22T + 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 + 4.78T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 4.74T + 97T^{2} \) |
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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
−9.013379712369381545310888876694, −7.74090915769233169104812180691, −7.50388159499122449336835681224, −6.28770825030048881813724462263, −6.05959889937136392211757443179, −4.88447829638734184787990485490, −3.94129943463316871707341075116, −3.49556610480522950858407703241, −3.03645639769226996550125614583, −1.16156701404506241434678792508,
1.16156701404506241434678792508, 3.03645639769226996550125614583, 3.49556610480522950858407703241, 3.94129943463316871707341075116, 4.88447829638734184787990485490, 6.05959889937136392211757443179, 6.28770825030048881813724462263, 7.50388159499122449336835681224, 7.74090915769233169104812180691, 9.013379712369381545310888876694