| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + 4.07i·7-s + i·8-s − 9-s − 3.28·11-s + i·12-s + 5.42i·13-s + 4.07·14-s + 16-s − 3.45i·17-s + i·18-s − 1.63·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.54i·7-s + 0.353i·8-s − 0.333·9-s − 0.990·11-s + 0.288i·12-s + 1.50i·13-s + 1.08·14-s + 0.250·16-s − 0.839i·17-s + 0.235i·18-s − 0.375·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4046233570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4046233570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.07iT - 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 - 5.42iT - 13T^{2} \) |
| 17 | \( 1 + 3.45iT - 17T^{2} \) |
| 19 | \( 1 + 1.63T + 19T^{2} \) |
| 23 | \( 1 + 0.611iT - 23T^{2} \) |
| 29 | \( 1 - 4.67T + 29T^{2} \) |
| 31 | \( 1 + 8.30T + 31T^{2} \) |
| 37 | \( 1 + 2.40iT - 37T^{2} \) |
| 41 | \( 1 - 2.93T + 41T^{2} \) |
| 43 | \( 1 + 8.64iT - 43T^{2} \) |
| 47 | \( 1 + 6.91iT - 47T^{2} \) |
| 53 | \( 1 - 6.79iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 + 9.06T + 71T^{2} \) |
| 73 | \( 1 + 9.10iT - 73T^{2} \) |
| 79 | \( 1 - 4.09T + 79T^{2} \) |
| 83 | \( 1 + 7.13iT - 83T^{2} \) |
| 89 | \( 1 + 4.80T + 89T^{2} \) |
| 97 | \( 1 - 14.7iT - 97T^{2} \) |
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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
−8.346500147786744842891568658591, −7.44733718030230057305529846176, −6.68001706270095953966654103867, −5.74293711048488200069759399263, −5.19524265096773538679726963218, −4.29890007350316657881543548212, −3.10083271593178617733922805199, −2.32661440727382460627171341159, −1.81936011343372177490417747524, −0.12871409434869098773246691635,
1.07323390893415443201854953779, 2.79330905768285801794535624537, 3.67201297497383202199901423118, 4.35660402892745833169424217230, 5.16832856960859075847346524434, 5.83188479946127051763272418927, 6.70740269506089982111900171635, 7.59291735655568843389205728554, 7.963893487884219500488241524160, 8.649803188004672525149608030435
