L(s) = 1 | + (2.53 − 4.53i)3-s + (3.53 + 10.6i)5-s + (23 − 23i)7-s + (−14.1 − 23.0i)9-s − 7.07i·11-s + (−24 − 24i)13-s + (57.0 + 10.8i)15-s + (77.7 + 77.7i)17-s + 68i·19-s + (−46 − 162. i)21-s + (−98.9 + 98.9i)23-s + (−100. + 75i)25-s + (−140. + 5.82i)27-s − 134.·29-s + 94·31-s + ⋯ |
L(s) = 1 | + (0.487 − 0.872i)3-s + (0.316 + 0.948i)5-s + (1.24 − 1.24i)7-s + (−0.523 − 0.851i)9-s − 0.193i·11-s + (−0.512 − 0.512i)13-s + (0.982 + 0.186i)15-s + (1.10 + 1.10i)17-s + 0.821i·19-s + (−0.478 − 1.68i)21-s + (−0.897 + 0.897i)23-s + (−0.800 + 0.599i)25-s + (−0.999 + 0.0415i)27-s − 0.860·29-s + 0.544·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.66989 - 0.649209i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66989 - 0.649209i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.53 + 4.53i)T \) |
| 5 | \( 1 + (-3.53 - 10.6i)T \) |
good | 7 | \( 1 + (-23 + 23i)T - 343iT^{2} \) |
| 11 | \( 1 + 7.07iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (24 + 24i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-77.7 - 77.7i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 - 68iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (98.9 - 98.9i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94T + 2.97e4T^{2} \) |
| 37 | \( 1 + (66 - 66i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 197. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-126 - 126i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-84.8 - 84.8i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (325. - 325. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 49.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 126T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-68 + 68i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 947. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-403 - 403i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 234iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-721. + 721. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-723 + 723i)T - 9.12e5iT^{2} \) |
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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
−14.28692401934880655825662920480, −13.68764790975247352797299396085, −12.26287024890642841933920940564, −10.99234233191301903774868918853, −9.981223914231789123555085394601, −7.972899572610735843763209526625, −7.48051906809941823493057201800, −5.90624856868930834177048412532, −3.61262379811086959444258348236, −1.62406629612410404771903496360,
2.27544583151828959995699602713, 4.63801260232617228321022465402, 5.43065947221431415231453552066, 7.952205393944289730752800745774, 8.928066350205377395405411070621, 9.765799578263983459595110383434, 11.41865875275496383406385969395, 12.31260881155533867703165569189, 13.92505231819335263054133457260, 14.70624245509862972501893785310