| L(s) = 1 | + (−2.16 + 9.50i)2-s + (3.54 − 4.44i)3-s + (−56.7 − 27.3i)4-s + (48.6 − 60.9i)5-s + (34.5 + 43.3i)6-s + (128. − 16.4i)7-s + (188. − 236. i)8-s + (46.8 + 205. i)9-s + (474. + 594. i)10-s + (137. − 603. i)11-s + (−323. + 155. i)12-s + (26.6 − 116. i)13-s + (−122. + 1.25e3i)14-s + (−98.7 − 432. i)15-s + (581. + 729. i)16-s + (−974. + 469. i)17-s + ⋯ |
| L(s) = 1 | + (−0.383 + 1.68i)2-s + (0.227 − 0.285i)3-s + (−1.77 − 0.854i)4-s + (0.869 − 1.09i)5-s + (0.392 + 0.491i)6-s + (0.991 − 0.127i)7-s + (1.04 − 1.30i)8-s + (0.192 + 0.845i)9-s + (1.49 + 1.88i)10-s + (0.343 − 1.50i)11-s + (−0.647 + 0.311i)12-s + (0.0437 − 0.191i)13-s + (−0.166 + 1.71i)14-s + (−0.113 − 0.496i)15-s + (0.567 + 0.711i)16-s + (−0.817 + 0.393i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.50028 + 0.781426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50028 + 0.781426i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-128. + 16.4i)T \) |
| good | 2 | \( 1 + (2.16 - 9.50i)T + (-28.8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (-3.54 + 4.44i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (-48.6 + 60.9i)T + (-695. - 3.04e3i)T^{2} \) |
| 11 | \( 1 + (-137. + 603. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-26.6 + 116. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (974. - 469. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 - 1.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-3.86e3 - 1.86e3i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 29 | \( 1 + (-1.11e3 + 536. i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 + 8.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.03e4 + 4.97e3i)T + (4.32e7 - 5.42e7i)T^{2} \) |
| 41 | \( 1 + (3.66e3 - 4.59e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (6.62e3 + 8.30e3i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (977. - 4.28e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.00e4 + 4.82e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-9.67e3 - 1.21e4i)T + (-1.59e8 + 6.96e8i)T^{2} \) |
| 61 | \( 1 + (2.91e4 - 1.40e4i)T + (5.26e8 - 6.60e8i)T^{2} \) |
| 67 | \( 1 + 2.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (2.41e4 + 1.16e4i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (1.07e4 + 4.70e4i)T + (-1.86e9 + 8.99e8i)T^{2} \) |
| 79 | \( 1 + 4.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-2.32e4 - 1.01e5i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (5.56e3 + 2.43e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 - 1.87e4T + 8.58e9T^{2} \) |
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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
−14.82968984779521214195700997726, −13.70410198854741017690169149704, −13.28998581407968487510367981267, −11.07116080964285280264141208721, −9.195147886937372297199024960826, −8.510101562047859367009412238232, −7.44173948475187167846937811213, −5.76366451298088654249921518068, −4.94638838587667402085443035768, −1.20463946638994245542349874541,
1.61302542277438407297879882807, 2.87631546896737626407775546176, 4.55892007568941649335757572426, 6.97668146720370095406576866809, 9.071295534564281959233485267448, 9.786307736001879710859096631331, 10.84171278591283763474294966787, 11.77024091864341498691461390467, 12.98717663635765523229962784231, 14.34351453449582813532608406459
