L(s) = 1 | + (−0.881 + 0.471i)2-s + (0.555 − 0.831i)4-s + (−0.995 + 0.0980i)7-s + (−0.0980 + 0.995i)8-s + (0.773 − 0.634i)9-s + (0.690 − 0.761i)11-s + (0.831 − 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (−0.249 + 0.997i)22-s + (−1.11 − 0.598i)23-s + (0.290 − 0.956i)25-s + (−0.471 + 0.881i)28-s + (−1.18 − 0.0584i)29-s + (0.773 + 0.634i)32-s + ⋯ |
L(s) = 1 | + (−0.881 + 0.471i)2-s + (0.555 − 0.831i)4-s + (−0.995 + 0.0980i)7-s + (−0.0980 + 0.995i)8-s + (0.773 − 0.634i)9-s + (0.690 − 0.761i)11-s + (0.831 − 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (−0.249 + 0.997i)22-s + (−1.11 − 0.598i)23-s + (0.290 − 0.956i)25-s + (−0.471 + 0.881i)28-s + (−1.18 − 0.0584i)29-s + (0.773 + 0.634i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6684353562\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6684353562\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.881 - 0.471i)T \) |
| 7 | \( 1 + (0.995 - 0.0980i)T \) |
good | 3 | \( 1 + (-0.773 + 0.634i)T^{2} \) |
| 5 | \( 1 + (-0.290 + 0.956i)T^{2} \) |
| 11 | \( 1 + (-0.690 + 0.761i)T + (-0.0980 - 0.995i)T^{2} \) |
| 13 | \( 1 + (0.956 - 0.290i)T^{2} \) |
| 17 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 19 | \( 1 + (0.471 - 0.881i)T^{2} \) |
| 23 | \( 1 + (1.11 + 0.598i)T + (0.555 + 0.831i)T^{2} \) |
| 29 | \( 1 + (1.18 + 0.0584i)T + (0.995 + 0.0980i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 0.326i)T + (0.881 - 0.471i)T^{2} \) |
| 41 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 43 | \( 1 + (-0.0330 + 0.0923i)T + (-0.773 - 0.634i)T^{2} \) |
| 47 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 53 | \( 1 + (-0.854 + 0.0419i)T + (0.995 - 0.0980i)T^{2} \) |
| 59 | \( 1 + (0.956 + 0.290i)T^{2} \) |
| 61 | \( 1 + (-0.634 - 0.773i)T^{2} \) |
| 67 | \( 1 + (-0.288 + 0.609i)T + (-0.634 - 0.773i)T^{2} \) |
| 71 | \( 1 + (-0.247 + 0.301i)T + (-0.195 - 0.980i)T^{2} \) |
| 73 | \( 1 + (-0.980 - 0.195i)T^{2} \) |
| 79 | \( 1 + (0.373 + 1.87i)T + (-0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.881 - 0.471i)T^{2} \) |
| 89 | \( 1 + (-0.555 + 0.831i)T^{2} \) |
| 97 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
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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
−9.324615331483747374014323384499, −8.749977844472073277488475334323, −7.82134438401865177216160418128, −6.97591421979249779094280069068, −6.25568899648079005687817178404, −5.86265755490593182806170244289, −4.39246016856384827674880773792, −3.43770488628147195144999332957, −2.17119638695024500666557284975, −0.71635883191683645333341498832,
1.41948079865956334238998529470, 2.41630847704097072546658755184, 3.63485762721097310819811212134, 4.26939670505984238820134804111, 5.68354571327936265913398230874, 6.73120425333733123524870542228, 7.27539419551318722740610071100, 7.959214767457915932477372983701, 9.043444402587169293966085871978, 9.707688482384232406541614277541