| L(s) = 1 | − 5-s − 11-s − 3·13-s − 2·17-s + 5·19-s − 8·23-s − 4·25-s + 7·29-s + 8·31-s − 3·37-s + 10·41-s − 10·43-s − 7·47-s − 2·53-s + 55-s + 9·59-s + 2·61-s + 3·65-s − 3·67-s + 6·71-s − 73-s + 10·79-s + 6·83-s + 2·85-s − 2·89-s − 5·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.832·13-s − 0.485·17-s + 1.14·19-s − 1.66·23-s − 4/5·25-s + 1.29·29-s + 1.43·31-s − 0.493·37-s + 1.56·41-s − 1.52·43-s − 1.02·47-s − 0.274·53-s + 0.134·55-s + 1.17·59-s + 0.256·61-s + 0.372·65-s − 0.366·67-s + 0.712·71-s − 0.117·73-s + 1.12·79-s + 0.658·83-s + 0.216·85-s − 0.211·89-s − 0.512·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.13300430167265, −14.52194342624957, −14.00986435551681, −13.59877415433610, −13.07748434897729, −12.15481917445668, −12.07245111767579, −11.58514530228557, −10.91923747435587, −10.16941363264009, −9.869502240977627, −9.419735909675984, −8.494028632169463, −8.046481134017760, −7.738617884741484, −6.942397363589959, −6.463050948238368, −5.791889844732914, −5.081962471374987, −4.590450174380606, −3.953458037655923, −3.243650492137786, −2.552002109300051, −1.916571047365813, −0.8696400786341645, 0,
0.8696400786341645, 1.916571047365813, 2.552002109300051, 3.243650492137786, 3.953458037655923, 4.590450174380606, 5.081962471374987, 5.791889844732914, 6.463050948238368, 6.942397363589959, 7.738617884741484, 8.046481134017760, 8.494028632169463, 9.419735909675984, 9.869502240977627, 10.16941363264009, 10.91923747435587, 11.58514530228557, 12.07245111767579, 12.15481917445668, 13.07748434897729, 13.59877415433610, 14.00986435551681, 14.52194342624957, 15.13300430167265
