| L(s) = 1 | + (2.49 + 0.733i)2-s + (−1.02 − 0.661i)4-s + (−2.23 + 15.5i)5-s + (−2.46 − 5.40i)7-s + (−15.7 − 18.1i)8-s + (−16.9 + 37.1i)10-s + (−20.9 + 6.16i)11-s + (−14.2 + 31.2i)13-s + (−2.20 − 15.3i)14-s + (−21.9 − 47.9i)16-s + (−108. + 69.9i)17-s + (−23.4 − 15.0i)19-s + (12.5 − 14.4i)20-s − 56.9·22-s + (18.9 − 108. i)23-s + ⋯ |
| L(s) = 1 | + (0.883 + 0.259i)2-s + (−0.128 − 0.0826i)4-s + (−0.199 + 1.38i)5-s + (−0.133 − 0.292i)7-s + (−0.694 − 0.801i)8-s + (−0.536 + 1.17i)10-s + (−0.575 + 0.168i)11-s + (−0.304 + 0.666i)13-s + (−0.0420 − 0.292i)14-s + (−0.342 − 0.749i)16-s + (−1.55 + 0.998i)17-s + (−0.283 − 0.182i)19-s + (0.140 − 0.162i)20-s − 0.551·22-s + (0.171 − 0.985i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0615720 + 0.821770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0615720 + 0.821770i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (-18.9 + 108. i)T \) |
| good | 2 | \( 1 + (-2.49 - 0.733i)T + (6.73 + 4.32i)T^{2} \) |
| 5 | \( 1 + (2.23 - 15.5i)T + (-119. - 35.2i)T^{2} \) |
| 7 | \( 1 + (2.46 + 5.40i)T + (-224. + 259. i)T^{2} \) |
| 11 | \( 1 + (20.9 - 6.16i)T + (1.11e3 - 719. i)T^{2} \) |
| 13 | \( 1 + (14.2 - 31.2i)T + (-1.43e3 - 1.66e3i)T^{2} \) |
| 17 | \( 1 + (108. - 69.9i)T + (2.04e3 - 4.46e3i)T^{2} \) |
| 19 | \( 1 + (23.4 + 15.0i)T + (2.84e3 + 6.23e3i)T^{2} \) |
| 29 | \( 1 + (10.5 - 6.78i)T + (1.01e4 - 2.21e4i)T^{2} \) |
| 31 | \( 1 + (10.3 + 11.9i)T + (-4.23e3 + 2.94e4i)T^{2} \) |
| 37 | \( 1 + (-33.6 - 233. i)T + (-4.86e4 + 1.42e4i)T^{2} \) |
| 41 | \( 1 + (53.6 - 373. i)T + (-6.61e4 - 1.94e4i)T^{2} \) |
| 43 | \( 1 + (310. - 358. i)T + (-1.13e4 - 7.86e4i)T^{2} \) |
| 47 | \( 1 - 547.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (192. + 421. i)T + (-9.74e4 + 1.12e5i)T^{2} \) |
| 59 | \( 1 + (-248. + 543. i)T + (-1.34e5 - 1.55e5i)T^{2} \) |
| 61 | \( 1 + (181. + 209. i)T + (-3.23e4 + 2.24e5i)T^{2} \) |
| 67 | \( 1 + (-164. - 48.3i)T + (2.53e5 + 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-218. - 64.1i)T + (3.01e5 + 1.93e5i)T^{2} \) |
| 73 | \( 1 + (-784. - 504. i)T + (1.61e5 + 3.53e5i)T^{2} \) |
| 79 | \( 1 + (-97.3 + 213. i)T + (-3.22e5 - 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-129. - 903. i)T + (-5.48e5 + 1.61e5i)T^{2} \) |
| 89 | \( 1 + (451. - 521. i)T + (-1.00e5 - 6.97e5i)T^{2} \) |
| 97 | \( 1 + (12.2 - 85.2i)T + (-8.75e5 - 2.57e5i)T^{2} \) |
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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
−12.71713458920080476710001066485, −11.39860422278458931688431433820, −10.58744048424511356363997795839, −9.657331375908098560902809235271, −8.280417143679537144760377566495, −6.69726803013860492445389621688, −6.53262959737475683866637148690, −4.84537531999829774691936748141, −3.84809716850029927674407023423, −2.53794509178180519312825509427,
0.24876697491820260084112867141, 2.46040118549546780864426915248, 3.97395222068252113825920835607, 5.00175497317268908813227067293, 5.65536569645910766338529738068, 7.48846055401008778057780717361, 8.730233329368495036676706803319, 9.158110181811802652505204463436, 10.77022067994987800065064568923, 11.93518885766461141540741579151
