| L(s) = 1 | − 1.67i·2-s + i·3-s − 0.806·4-s + (−1.48 + 1.67i)5-s + 1.67·6-s − i·7-s − 1.99i·8-s − 9-s + (2.80 + 2.48i)10-s + 11-s − 0.806i·12-s − 4.48i·13-s − 1.67·14-s + (−1.67 − 1.48i)15-s − 4.96·16-s + 3.15i·17-s + ⋯ |
| L(s) = 1 | − 1.18i·2-s + 0.577i·3-s − 0.403·4-s + (−0.662 + 0.749i)5-s + 0.683·6-s − 0.377i·7-s − 0.707i·8-s − 0.333·9-s + (0.887 + 0.784i)10-s + 0.301·11-s − 0.232i·12-s − 1.24i·13-s − 0.447·14-s + (−0.432 − 0.382i)15-s − 1.24·16-s + 0.765i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.226175136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226175136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.48 - 1.67i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.67iT - 2T^{2} \) |
| 13 | \( 1 + 4.48iT - 13T^{2} \) |
| 17 | \( 1 - 3.15iT - 17T^{2} \) |
| 19 | \( 1 + 0.350T + 19T^{2} \) |
| 23 | \( 1 + 2.19iT - 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 + 0.0630T + 31T^{2} \) |
| 37 | \( 1 + 9.44iT - 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 + 7.79iT - 43T^{2} \) |
| 47 | \( 1 + 3.98iT - 47T^{2} \) |
| 53 | \( 1 + 7.31iT - 53T^{2} \) |
| 59 | \( 1 - 5.24T + 59T^{2} \) |
| 61 | \( 1 + 7.73T + 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 1.13T + 71T^{2} \) |
| 73 | \( 1 + 0.649iT - 73T^{2} \) |
| 79 | \( 1 - 3.93T + 79T^{2} \) |
| 83 | \( 1 + 4.19iT - 83T^{2} \) |
| 89 | \( 1 - 3.35T + 89T^{2} \) |
| 97 | \( 1 - 2.64iT - 97T^{2} \) |
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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
−10.01020183055042510675645122612, −8.829635011345206445648910475127, −8.002655617579232838190624911339, −7.00079183137110927287931377478, −6.17689686594975981113137661917, −4.81613380021400245053599645262, −3.78560684874775231478740119355, −3.31972430420178243043930732286, −2.25715316595668248819791991148, −0.55258995261200730074051103139,
1.42868309678974794188713625687, 2.86087897167701832064282874163, 4.43332265711404926463108589245, 5.05353503070043138312815769339, 6.16312310168173173138716210881, 6.79486259299016473541371040838, 7.53178376628211756894497240179, 8.365073765459515972652577679006, 8.831087986546491816433415701702, 9.720042890646207520486679083166
