| L(s) = 1 | − 0.706·3-s − 1.32i·5-s − 1.54·7-s − 2.50·9-s + 6.14·11-s − 4.11i·13-s + 0.933i·15-s + 6.36i·17-s + 1.27i·19-s + 1.09·21-s + 3.80i·23-s + 3.25·25-s + 3.88·27-s + 8.80i·29-s − 7.10i·31-s + ⋯ |
| L(s) = 1 | − 0.408·3-s − 0.590i·5-s − 0.585·7-s − 0.833·9-s + 1.85·11-s − 1.14i·13-s + 0.241i·15-s + 1.54i·17-s + 0.292i·19-s + 0.238·21-s + 0.792i·23-s + 0.650·25-s + 0.748·27-s + 1.63i·29-s − 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.374531644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374531644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (4.43 + 4.16i)T \) |
| good | 3 | \( 1 + 0.706T + 3T^{2} \) |
| 5 | \( 1 + 1.32iT - 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 - 6.14T + 11T^{2} \) |
| 13 | \( 1 + 4.11iT - 13T^{2} \) |
| 17 | \( 1 - 6.36iT - 17T^{2} \) |
| 19 | \( 1 - 1.27iT - 19T^{2} \) |
| 23 | \( 1 - 3.80iT - 23T^{2} \) |
| 29 | \( 1 - 8.80iT - 29T^{2} \) |
| 31 | \( 1 + 7.10iT - 31T^{2} \) |
| 41 | \( 1 + 3.93T + 41T^{2} \) |
| 43 | \( 1 + 9.78iT - 43T^{2} \) |
| 47 | \( 1 - 8.54T + 47T^{2} \) |
| 53 | \( 1 - 6.19T + 53T^{2} \) |
| 59 | \( 1 + 13.3iT - 59T^{2} \) |
| 61 | \( 1 - 2.10iT - 61T^{2} \) |
| 67 | \( 1 - 2.10T + 67T^{2} \) |
| 71 | \( 1 - 6.83T + 71T^{2} \) |
| 73 | \( 1 - 6.78T + 73T^{2} \) |
| 79 | \( 1 - 0.376iT - 79T^{2} \) |
| 83 | \( 1 - 8.15T + 83T^{2} \) |
| 89 | \( 1 + 11.8iT - 89T^{2} \) |
| 97 | \( 1 + 8.18iT - 97T^{2} \) |
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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
−8.786716556295524710807190442077, −8.399397842528210805264128376130, −7.23811504720832249314634608898, −6.40181728434858098268168418049, −5.78901455809970099108241088864, −5.11403370148822665580119860475, −3.81658518270444208653528414837, −3.39975247382477424582258049648, −1.81772612982557853009869887858, −0.67236353652834728647983531562,
0.909352956159707766839162391332, 2.43108666915312061382296788177, 3.26768196501562869291946529144, 4.25946300556295876100944865071, 5.08936198359595867935233249015, 6.38674987129424046268442507152, 6.51673929929556299828726810859, 7.20659675489868175080265420396, 8.517318897281905225068618913788, 9.146004578353291426463033909409
