| L(s) = 1 | − 1.44i·2-s + 2.24i·3-s − 0.0881·4-s + 3.24·6-s − 1.35i·7-s − 2.76i·8-s − 2.04·9-s + 4.85·11-s − 0.198i·12-s + 0.198i·13-s − 1.96·14-s − 4.16·16-s + 1.13i·17-s + 2.96i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 1.02i·2-s + 1.29i·3-s − 0.0440·4-s + 1.32·6-s − 0.512i·7-s − 0.976i·8-s − 0.682·9-s + 1.46·11-s − 0.0571i·12-s + 0.0549i·13-s − 0.524·14-s − 1.04·16-s + 0.275i·17-s + 0.697i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65055 - 0.389642i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65055 - 0.389642i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.44iT - 2T^{2} \) |
| 3 | \( 1 - 2.24iT - 3T^{2} \) |
| 7 | \( 1 + 1.35iT - 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 13 | \( 1 - 0.198iT - 13T^{2} \) |
| 17 | \( 1 - 1.13iT - 17T^{2} \) |
| 23 | \( 1 - 2.55iT - 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 - 0.137iT - 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 7.59iT - 43T^{2} \) |
| 47 | \( 1 + 2.69iT - 47T^{2} \) |
| 53 | \( 1 - 12.8iT - 53T^{2} \) |
| 59 | \( 1 + 5.82T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 + 8.01iT - 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 + 11.9iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.77iT - 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 + 0.198iT - 97T^{2} \) |
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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
−10.66983261491630337398983370990, −10.31423982737219632265130438588, −9.475553870652106182648740403748, −8.752126304298894193957437230188, −7.19337332813227344787833958691, −6.25104032855813785858033513801, −4.69153927370954208362856479753, −3.93482387184522929585060695616, −3.13951272370970562473039845349, −1.38846569744502568888483710772,
1.44445002251482264102709960173, 2.77346727912974107959307975080, 4.64889387766109207343427901106, 5.95757399890772502841962673460, 6.60324887940402443710971776633, 7.13124052512751507702271868084, 8.230359069998112799026925103916, 8.751345546340807271148216303619, 10.05335181675510819254702750952, 11.56695959164960302141611259973
