| 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 | \( 1 + (-137. + 957. i)T + (-8.75e5 - 2.57e5i)T^{2} \) |
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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
−12.38467448505142873394293427648, −11.18306548481612108138153111023, −10.39828241923784854272508351071, −9.383198324518936807120003405841, −8.121366925224727110718254042151, −7.09480419163690443514612589154, −6.38872145182474518572896453079, −3.92109163956220838060268834384, −2.68647706534103439684981814800, −0.68818276105916657816620436303,
1.71059317796778733719149158664, 3.64730823601408280769115121414, 5.28422548975479005348230843986, 6.46813701569707791177856504599, 8.105576283313937236096024341651, 8.874322674183888022437616655186, 9.547230003511834087322613608389, 10.70168703687215468974434646025, 12.16207952158336029649739870992, 12.62759613402357849546453572973
