| L(s) = 1 | + i·3-s + 7-s − i·11-s + (−1 + i)17-s + (−1 + i)19-s + i·21-s + (1 − i)23-s + i·25-s + i·27-s + (−1 − i)29-s + 33-s − i·37-s − i·41-s − 47-s + (−1 − i)51-s + ⋯ |
| L(s) = 1 | + i·3-s + 7-s − i·11-s + (−1 + i)17-s + (−1 + i)19-s + i·21-s + (1 − i)23-s + i·25-s + i·27-s + (−1 − i)29-s + 33-s − i·37-s − i·41-s − 47-s + (−1 − i)51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9947977129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9947977129\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (-1 + i)T - iT^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (1 + i)T + iT^{2} \) |
| 97 | \( 1 - iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80425694797480094780817697818, −10.45285528758774025243278030759, −9.138532724907290560430771113356, −8.603138559915031714660163110738, −7.65046914301602076641653548517, −6.35059854574627329617751132701, −5.36340343605675624584011325463, −4.36647037735321642176788376480, −3.63925105812743028936809842145, −1.92706715502051711279701527043,
1.53244685606901760957159962672, 2.54904202426181270923106638273, 4.41705991021008761778126021756, 5.08956598877109141355075072311, 6.66276018649629406321470989938, 7.09386504988478485409094998737, 7.994376404226445290662460566884, 8.901897342858475123168183115579, 9.857369490362876240893243708340, 11.06360576595109667388945836977
