L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + (−1.07 − 1.95i)5-s + 1.00·6-s + (3.43 + 3.43i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (2.14 + 0.624i)10-s − 1.77i·11-s + (−0.707 + 0.707i)12-s + (1.21 + 1.21i)13-s − 4.85·14-s + (−0.624 + 2.14i)15-s − 1.00·16-s + (−1.12 − 1.12i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (−0.481 − 0.876i)5-s + 0.408·6-s + (1.29 + 1.29i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.678 + 0.197i)10-s − 0.536i·11-s + (−0.204 + 0.204i)12-s + (0.337 + 0.337i)13-s − 1.29·14-s + (−0.161 + 0.554i)15-s − 0.250·16-s + (−0.272 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07105 + 0.0296699i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07105 + 0.0296699i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.07 + 1.95i)T \) |
| 23 | \( 1 + (-4.76 + 0.510i)T \) |
good | 7 | \( 1 + (-3.43 - 3.43i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.77iT - 11T^{2} \) |
| 13 | \( 1 + (-1.21 - 1.21i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.12 + 1.12i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 29 | \( 1 - 3.23iT - 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 + (-0.321 - 0.321i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + (-6.47 + 6.47i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9.10 + 9.10i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.42 + 6.42i)T - 53iT^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 + 14.4iT - 61T^{2} \) |
| 67 | \( 1 + (-0.586 - 0.586i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + (-3.84 - 3.84i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.55T + 79T^{2} \) |
| 83 | \( 1 + (2.56 - 2.56i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.44T + 89T^{2} \) |
| 97 | \( 1 + (-7.79 - 7.79i)T + 97iT^{2} \) |
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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
−10.65755125203145263528157560765, −9.070465795609330425207189420360, −8.731914649173048843811457106418, −8.055318787790011873865039212837, −7.09328738143492498042550881665, −5.88302823852767019684406375241, −5.25316657376776991571476171507, −4.36548876603513812534289775751, −2.30497590542908274437962396877, −1.00689617252069077889543979487,
1.03475265643348483738321582965, 2.67308099811598484422459727799, 4.12978247369033949632575904558, 4.47531549852376221533279790218, 6.11245411389082181553897142635, 7.29278829321260618076913263324, 7.72510649938449479924386909403, 8.751854153696338708832276329461, 9.962033917815709290879404029338, 10.69777159863644583719850966530