| L(s) = 1 | − i·2-s + 3-s − 4-s − i·6-s − 3i·7-s + i·8-s − 2·9-s − 12-s + (−2 − 3i)13-s − 3·14-s + 16-s − 3·17-s + 2i·18-s − 6i·19-s − 3i·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.408i·6-s − 1.13i·7-s + 0.353i·8-s − 0.666·9-s − 0.288·12-s + (−0.554 − 0.832i)13-s − 0.801·14-s + 0.250·16-s − 0.727·17-s + 0.471i·18-s − 1.37i·19-s − 0.654i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.364717 - 1.20457i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364717 - 1.20457i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2 + 3i)T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 15iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−10.34971898797595989320252739846, −9.308198244815035831406713849316, −8.682234634856913125841328137352, −7.64929583800892230370343949942, −6.85785597870050714067986650135, −5.37673738251340324102294189919, −4.41261918207217047788383709451, −3.29042029706112970569274377792, −2.43288778495280912154802728314, −0.61761418858325786133549348357,
2.10380209540382038011584865571, 3.24778115972941505072309752789, 4.59344199134021636518576752244, 5.60108455906016923837017513258, 6.39822660865749472858892430044, 7.47651389342887512601851415314, 8.407206674963692957500719941792, 9.002007262895981924125233786835, 9.605776554049539960768405289056, 10.89329599547418811356503150106
