L(s) = 1 | + (2.71 + 2.71i)2-s + (16.9 − 16.9i)3-s − 49.2i·4-s + 92.2·6-s + (−383. − 383. i)7-s + (307. − 307. i)8-s + 152. i·9-s + 2.40e3·11-s + (−836. − 836. i)12-s + (332. − 332. i)13-s − 2.08e3i·14-s − 1.48e3·16-s + (373. + 373. i)17-s + (−412. + 412. i)18-s + 7.03e3i·19-s + ⋯ |
L(s) = 1 | + (0.339 + 0.339i)2-s + (0.629 − 0.629i)3-s − 0.769i·4-s + 0.427·6-s + (−1.11 − 1.11i)7-s + (0.600 − 0.600i)8-s + 0.208i·9-s + 1.80·11-s + (−0.484 − 0.484i)12-s + (0.151 − 0.151i)13-s − 0.759i·14-s − 0.361·16-s + (0.0760 + 0.0760i)17-s + (−0.0707 + 0.0707i)18-s + 1.02i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.80689 - 1.13019i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80689 - 1.13019i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-2.71 - 2.71i)T + 64iT^{2} \) |
| 3 | \( 1 + (-16.9 + 16.9i)T - 729iT^{2} \) |
| 7 | \( 1 + (383. + 383. i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 - 2.40e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-332. + 332. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (-373. - 373. i)T + 2.41e7iT^{2} \) |
| 19 | \( 1 - 7.03e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (2.61e3 - 2.61e3i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 - 1.36e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.07e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + (2.99e4 + 2.99e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 6.03e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + (-6.38e4 + 6.38e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + (-7.43e4 - 7.43e4i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + (1.25e5 - 1.25e5i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + 1.52e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.34e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + (-6.90e4 - 6.90e4i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 + 9.61e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-2.31e5 + 2.31e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 - 2.81e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (5.49e5 - 5.49e5i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + 1.19e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-4.69e5 - 4.69e5i)T + 8.32e11iT^{2} \) |
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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
−16.00608209142159549213048590359, −14.31969620660356902009280880334, −13.89383253724418506064865350265, −12.61804192393305451622410238432, −10.60023559490217574932361444881, −9.317263821266561506861559010442, −7.32345295788439426667361550663, −6.25571370892031099511169605180, −3.89013370680782036292210412741, −1.25165173809262749239888439028,
2.84724527591889659009133171278, 4.06497223377640138429993403034, 6.52225920956419586112800239273, 8.759879072773892671711783856103, 9.478753704732632725781467110255, 11.64176822172687406453174680904, 12.52802158882921132621272852878, 13.95930869155574276442624856711, 15.24211861614758397958175714458, 16.30023501451669856152724518831