L(s) = 1 | + 4i·2-s − 16·4-s + 233i·7-s − 64i·8-s + 498·11-s − 809i·13-s − 932·14-s + 256·16-s + 1.00e3i·17-s + 1.70e3·19-s + 1.99e3i·22-s + 1.55e3i·23-s + 3.23e3·26-s − 3.72e3i·28-s + 7.83e3·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.79i·7-s − 0.353i·8-s + 1.24·11-s − 1.32i·13-s − 1.27·14-s + 0.250·16-s + 0.840i·17-s + 1.08·19-s + 0.877i·22-s + 0.612i·23-s + 0.938·26-s − 0.898i·28-s + 1.72·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.284518836\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.284518836\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 233iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 498T + 1.61e5T^{2} \) |
| 13 | \( 1 + 809iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.00e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.55e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 7.83e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 977T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.82e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 8.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.94e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.41e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.76e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.52e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.74e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.54e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 646iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.15e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 7.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.45e4iT - 8.58e9T^{2} \) |
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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
−10.39327360418251247698837314349, −9.394012638392848689329039917442, −8.707883497582276091591723809566, −7.982534451473953299366396006029, −6.74461202560975420272404410661, −5.78559822479078762602279715531, −5.28204674117611898410969703455, −3.77961938066780321635110940594, −2.60053570576630696783709110047, −1.05999910540025849415515964428,
0.70177569216353709191054009788, 1.42312071288491152235082666376, 3.05843028153110295655057029766, 4.16231082955827487946164769186, 4.67852629915443502086391780684, 6.51142085319768684907579093637, 7.10403391905505918909641437101, 8.266681119550813182480584871326, 9.481658581628181040781009738857, 9.894035240479646785678045928823