| L(s) = 1 | + (6.26 + 2.27i)3-s + (2.15 + 12.2i)5-s + (−4.57 + 7.92i)7-s + (13.3 + 11.1i)9-s + (−7.90 − 13.6i)11-s + (23.5 − 8.57i)13-s + (−14.3 + 81.3i)15-s + (23.2 − 19.4i)17-s + (82.6 + 5.43i)19-s + (−46.7 + 39.1i)21-s + (−2.18 + 12.4i)23-s + (−27.0 + 9.82i)25-s + (−32.0 − 55.4i)27-s + (−78.9 − 66.2i)29-s + (117. − 203. i)31-s + ⋯ |
| L(s) = 1 | + (1.20 + 0.438i)3-s + (0.192 + 1.09i)5-s + (−0.247 + 0.427i)7-s + (0.493 + 0.413i)9-s + (−0.216 − 0.375i)11-s + (0.502 − 0.182i)13-s + (−0.246 + 1.40i)15-s + (0.331 − 0.278i)17-s + (0.997 + 0.0656i)19-s + (−0.485 + 0.407i)21-s + (−0.0198 + 0.112i)23-s + (−0.216 + 0.0786i)25-s + (−0.228 − 0.395i)27-s + (−0.505 − 0.424i)29-s + (0.680 − 1.17i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.85597 + 0.973744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85597 + 0.973744i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-82.6 - 5.43i)T \) |
| good | 3 | \( 1 + (-6.26 - 2.27i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-2.15 - 12.2i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (4.57 - 7.92i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (7.90 + 13.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-23.5 + 8.57i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-23.2 + 19.4i)T + (853. - 4.83e3i)T^{2} \) |
| 23 | \( 1 + (2.18 - 12.4i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (78.9 + 66.2i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-117. + 203. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (361. + 131. i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (83.3 + 472. i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-412. - 346. i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + (93.1 - 528. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-305. + 256. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (13.1 - 74.6i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (160. + 134. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-134. - 764. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + (826. + 300. i)T + (2.98e5 + 2.50e5i)T^{2} \) |
| 79 | \( 1 + (-539. - 196. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (472. - 817. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-127. + 46.4i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-525. + 440. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−14.09256758647078912885582238813, −13.62647248615126995448385159921, −11.94082878883475173317533671627, −10.61862593523988166780978969498, −9.629939779002673621701175199322, −8.571957224279723585423561062937, −7.32530304698479705804807518486, −5.79087414403627693516498863040, −3.59630084837790586094518870730, −2.62719000830198480146097932334,
1.49096837130701675303792983073, 3.39342077007562257147807173412, 5.12712682167275536038398646374, 7.03186489089175250198751114481, 8.264323037785172934541964878760, 9.029580084474802527960463978739, 10.21254332023120708513080002169, 11.97547652630377818281958448648, 13.09472486983710461286957623473, 13.64632852573378289411839104005
