| L(s) = 1 | + (−1.91 + 0.563i)2-s + (1.96 + 2.26i)3-s + (3.36 − 2.16i)4-s + (−1.40 − 9.79i)5-s + (−5.04 − 3.24i)6-s + (−9.01 + 19.7i)7-s + (−5.23 + 6.04i)8-s + (−1.28 + 8.90i)9-s + (8.22 + 18.0i)10-s + (28.1 + 8.26i)11-s + (11.5 + 3.38i)12-s + (7.25 + 15.8i)13-s + (6.17 − 42.9i)14-s + (19.4 − 22.4i)15-s + (6.64 − 14.5i)16-s + (74.9 + 48.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.126 − 0.876i)5-s + (−0.343 − 0.220i)6-s + (−0.486 + 1.06i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.260 + 0.569i)10-s + (0.771 + 0.226i)11-s + (0.276 + 0.0813i)12-s + (0.154 + 0.338i)13-s + (0.117 − 0.819i)14-s + (0.334 − 0.386i)15-s + (0.103 − 0.227i)16-s + (1.06 + 0.687i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.741046 + 0.835516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.741046 + 0.835516i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.91 - 0.563i)T \) |
| 3 | \( 1 + (-1.96 - 2.26i)T \) |
| 23 | \( 1 + (31.0 - 105. i)T \) |
| good | 5 | \( 1 + (1.40 + 9.79i)T + (-119. + 35.2i)T^{2} \) |
| 7 | \( 1 + (9.01 - 19.7i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (-28.1 - 8.26i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (-7.25 - 15.8i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (-74.9 - 48.1i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (109. - 70.2i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (-131. - 84.4i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (211. - 244. i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (-20.8 + 145. i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (8.64 + 60.1i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-89.6 - 103. i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 + 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-206. + 452. i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (232. + 509. i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (275. - 318. i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (-527. + 154. i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-597. + 175. i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (-695. + 447. i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (-210. - 461. i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-182. + 1.27e3i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (-407. - 470. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-137. - 957. i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.62759613402357849546453572973, −12.16207952158336029649739870992, −10.70168703687215468974434646025, −9.547230003511834087322613608389, −8.874322674183888022437616655186, −8.105576283313937236096024341651, −6.46813701569707791177856504599, −5.28422548975479005348230843986, −3.64730823601408280769115121414, −1.71059317796778733719149158664,
0.68818276105916657816620436303, 2.68647706534103439684981814800, 3.92109163956220838060268834384, 6.38872145182474518572896453079, 7.09480419163690443514612589154, 8.121366925224727110718254042151, 9.383198324518936807120003405841, 10.39828241923784854272508351071, 11.18306548481612108138153111023, 12.38467448505142873394293427648
