| L(s) = 1 | + (−1 − i)2-s + 2i·4-s + (−2 + i)5-s + (2 − 2i)8-s + 3i·9-s + (3 + i)10-s + (1 − i)13-s − 4·16-s + (−3 − 3i)17-s + (3 − 3i)18-s + (−2 − 4i)20-s + (3 − 4i)25-s − 2·26-s − 10·29-s + (4 + 4i)32-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s + (−0.894 + 0.447i)5-s + (0.707 − 0.707i)8-s + i·9-s + (0.948 + 0.316i)10-s + (0.277 − 0.277i)13-s − 16-s + (−0.727 − 0.727i)17-s + (0.707 − 0.707i)18-s + (−0.447 − 0.894i)20-s + (0.600 − 0.800i)25-s − 0.392·26-s − 1.85·29-s + (0.707 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (1 + i)T \) |
| 5 | \( 1 + (2 - i)T \) |
| 37 | \( 1 + (6 + i)T \) |
| good | 3 | \( 1 - 3iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (3 + 3i)T + 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (5 + 5i)T + 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (-13 - 13i)T + 97iT^{2} \) |
| 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
−10.10428359662787337066770518094, −9.016375028131753396803122422610, −8.254932627667305479156273850089, −7.48888235165029027821752309908, −6.82484764981506984045492994753, −5.18554313264763678897389366079, −4.07867576305383418033703019225, −3.11599716682158948165795648927, −1.95863352292669749078152531455, 0,
1.57021912013000572876739886236, 3.57226654403372571652121585562, 4.54769401646004883654516575513, 5.70874552038808586344125287731, 6.63858501741340341941558566839, 7.40247154014426288428788206943, 8.371720403993575076604923550551, 8.951394856059769887022274884015, 9.680603197913957584464900716548
