L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.707 + 0.707i)3-s + (−0.499 − 0.866i)4-s + (0.258 + 0.965i)6-s − 0.999·8-s − 1.00i·9-s − 1.73·11-s + (0.965 + 0.258i)12-s + (−0.5 + 0.866i)16-s + (−0.258 + 0.448i)17-s + (−0.866 − 0.500i)18-s + (0.965 + 1.67i)19-s + (−0.866 + 1.49i)22-s + (0.707 − 0.707i)24-s + 25-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.707 + 0.707i)3-s + (−0.499 − 0.866i)4-s + (0.258 + 0.965i)6-s − 0.999·8-s − 1.00i·9-s − 1.73·11-s + (0.965 + 0.258i)12-s + (−0.5 + 0.866i)16-s + (−0.258 + 0.448i)17-s + (−0.866 − 0.500i)18-s + (0.965 + 1.67i)19-s + (−0.866 + 1.49i)22-s + (0.707 − 0.707i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8405555005\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8405555005\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.048828873211688569752095126461, −8.220582860894971811864373042028, −7.29327890720426845162985312079, −6.08667979084908883665918674627, −5.63497173758640337364383036845, −4.96658710903795889361629479691, −4.21713991038606427900799625922, −3.34562524989712566389833601803, −2.56479857656680580305347198426, −1.17655766269096546513777339902,
0.50594352422480360120118069525, 2.42497148962263169970410720202, 3.05051668792833283921210866630, 4.53181737911393061870196293912, 5.14173382041819264115829695606, 5.54666598920519380992862880227, 6.55073251217180823029943831627, 7.21161834522985758169414083167, 7.59700834196780211113916915432, 8.433409183250190724559903755492