| L(s) = 1 | − i·4-s + 5-s + (1 − i)7-s − i·9-s + i·11-s − 16-s + (−1 + i)17-s − 19-s − i·20-s + (−1 + i)23-s + 25-s + (−1 − i)28-s + (1 − i)35-s − 36-s + (1 + i)43-s + 44-s + ⋯ |
| L(s) = 1 | − i·4-s + 5-s + (1 − i)7-s − i·9-s + i·11-s − 16-s + (−1 + i)17-s − 19-s − i·20-s + (−1 + i)23-s + 25-s + (−1 − i)28-s + (1 − i)35-s − 36-s + (1 + i)43-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.257709130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.257709130\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - iT \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + iT^{2} \) |
| 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + (-1 - i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.928992766231498495283202157915, −9.434508229280321716747687049869, −8.472694984801156329706877199512, −7.31185976698375435565366289992, −6.45161028283209600610327061817, −5.86651309342698863519271881083, −4.68333793336964788747238850670, −4.09145438720171200006255364458, −2.13086677055753496932109963622, −1.38647406043806404452258546928,
2.24463114706879205074679729136, 2.49991502080536803072676393644, 4.21119976371123900749858616250, 5.09481923454441098444602380516, 5.91177030800592252780109755831, 6.92250198409690946150153558808, 8.001380704027362578699331707722, 8.677848217376404455181757425027, 9.022136469844090951435148453823, 10.44199131614385292263510131788
