L(s) = 1 | + (0.955 − 0.294i)2-s + (0.266 + 0.680i)3-s + (0.826 − 0.563i)4-s + (0.455 + 0.571i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (0.341 − 0.317i)9-s + (−1.40 − 1.29i)11-s + (0.603 + 0.411i)12-s + (−0.425 + 1.86i)13-s + (0.365 − 0.930i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.233 − 0.403i)18-s + (0.698 + 0.215i)21-s + (−1.72 − 0.829i)22-s + ⋯ |
L(s) = 1 | + (0.955 − 0.294i)2-s + (0.266 + 0.680i)3-s + (0.826 − 0.563i)4-s + (0.455 + 0.571i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (0.341 − 0.317i)9-s + (−1.40 − 1.29i)11-s + (0.603 + 0.411i)12-s + (−0.425 + 1.86i)13-s + (0.365 − 0.930i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.233 − 0.403i)18-s + (0.698 + 0.215i)21-s + (−1.72 − 0.829i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.620364540\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.620364540\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.955 + 0.294i)T \) |
| 7 | \( 1 + (-0.623 + 0.781i)T \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T \) |
good | 3 | \( 1 + (-0.266 - 0.680i)T + (-0.733 + 0.680i)T^{2} \) |
| 5 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 11 | \( 1 + (1.40 + 1.29i)T + (0.0747 + 0.997i)T^{2} \) |
| 13 | \( 1 + (0.425 - 1.86i)T + (-0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.0332 + 0.443i)T + (-0.988 + 0.149i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 53 | \( 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 61 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.78 - 0.858i)T + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 79 | \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (0.535 - 0.496i)T + (0.0747 - 0.997i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.876694023626695280628366363633, −7.87795270680791736355432575844, −7.02970575319381856498580830922, −6.55397446265531249749594981385, −5.25161548501208230034663628553, −4.85654544998407510798183896170, −4.11342964533333174740743259444, −3.31719893358087850546903761491, −2.49656614761230653677973037554, −1.20486968366875468888792793000,
1.80198474382053369787654802187, 2.40181367480797648503921466942, 3.19075787722016316452044159530, 4.53396780064768752285200825279, 5.13793909460570075653949996881, 5.65056222054682167401128699347, 6.65081486020073554243824323791, 7.59832085548630627620107521337, 7.87937342393283821640909186136, 8.310031462084086188255966356080