| L(s) = 1 | − 1.78i·2-s − 1.19·4-s + (−0.766 − 2.10i)5-s − 4.04i·7-s − 1.43i·8-s + (−3.75 + 1.37i)10-s − 4.80·11-s − 0.533i·13-s − 7.23·14-s − 4.96·16-s + 0.299i·17-s + 6.02·19-s + (0.919 + 2.51i)20-s + 8.60i·22-s + 0.379i·23-s + ⋯ |
| L(s) = 1 | − 1.26i·2-s − 0.599·4-s + (−0.343 − 0.939i)5-s − 1.52i·7-s − 0.506i·8-s + (−1.18 + 0.433i)10-s − 1.45·11-s − 0.147i·13-s − 1.93·14-s − 1.24·16-s + 0.0727i·17-s + 1.38·19-s + (0.205 + 0.562i)20-s + 1.83i·22-s + 0.0790i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107983054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107983054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.766 + 2.10i)T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.78iT - 2T^{2} \) |
| 7 | \( 1 + 4.04iT - 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 + 0.533iT - 13T^{2} \) |
| 17 | \( 1 - 0.299iT - 17T^{2} \) |
| 19 | \( 1 - 6.02T + 19T^{2} \) |
| 23 | \( 1 - 0.379iT - 23T^{2} \) |
| 31 | \( 1 - 9.14T + 31T^{2} \) |
| 37 | \( 1 - 8.51iT - 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 6.01iT - 43T^{2} \) |
| 47 | \( 1 - 0.299iT - 47T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 - 6.53T + 59T^{2} \) |
| 61 | \( 1 + 0.755T + 61T^{2} \) |
| 67 | \( 1 - 3.95iT - 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 11.0iT - 73T^{2} \) |
| 79 | \( 1 + 7.01T + 79T^{2} \) |
| 83 | \( 1 - 6.15iT - 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 2.41iT - 97T^{2} \) |
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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
−9.453891846731599843073264632871, −8.245273871899356241990064744584, −7.65381989793912506011037606710, −6.79252585603308216238969806641, −5.31684442405100020232063069426, −4.56013035389439545327072238029, −3.69876379422601155808981869859, −2.81346818761928824551722964537, −1.35850127342481252984351475844, −0.47359066027052097360828361037,
2.42216914732772039065148939808, 2.94554163812445357247440289525, 4.62699895270866204621536812519, 5.60861176816376885849641432905, 5.95050654104983551123568788744, 7.02403908294422874310325666958, 7.69570593691617922473638415990, 8.271045332352554611730048091563, 9.157266186966739338018185462778, 10.06324039107353842343374842242
