L(s) = 1 | + 2-s + (1.72 + 0.163i)3-s + 4-s − 1.55i·5-s + (1.72 + 0.163i)6-s − i·7-s + 8-s + (2.94 + 0.565i)9-s − 1.55i·10-s + (−0.560 + 3.26i)11-s + (1.72 + 0.163i)12-s − 2.32i·13-s − i·14-s + (0.254 − 2.68i)15-s + 16-s − 6.13·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.995 + 0.0946i)3-s + 0.5·4-s − 0.695i·5-s + (0.703 + 0.0669i)6-s − 0.377i·7-s + 0.353·8-s + (0.982 + 0.188i)9-s − 0.491i·10-s + (−0.168 + 0.985i)11-s + (0.497 + 0.0473i)12-s − 0.645i·13-s − 0.267i·14-s + (0.0658 − 0.692i)15-s + 0.250·16-s − 1.48·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.82416 - 0.375782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.82416 - 0.375782i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.72 - 0.163i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (0.560 - 3.26i)T \) |
good | 5 | \( 1 + 1.55iT - 5T^{2} \) |
| 13 | \( 1 + 2.32iT - 13T^{2} \) |
| 17 | \( 1 + 6.13T + 17T^{2} \) |
| 19 | \( 1 - 3.69iT - 19T^{2} \) |
| 23 | \( 1 + 4.25iT - 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 - 5.48iT - 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 6.56iT - 59T^{2} \) |
| 61 | \( 1 + 7.35iT - 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 + 2.88iT - 71T^{2} \) |
| 73 | \( 1 + 1.67iT - 73T^{2} \) |
| 79 | \( 1 + 12.0iT - 79T^{2} \) |
| 83 | \( 1 - 0.338T + 83T^{2} \) |
| 89 | \( 1 + 5.55iT - 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−10.90197700115210422403558015041, −10.15370895357502765478978419574, −9.148760085129283132101748847824, −8.258717556612482332290814183397, −7.39543647477026918376914719372, −6.40028507863242988479972011885, −4.83878599560099243218866008337, −4.33366881712540859763460666700, −3.01893115730309873547401585312, −1.74866350235044795566299166611,
2.09664687407650192185266966206, 3.05479973795611846248110953746, 4.02809446168922998692987682305, 5.30606830742799958889673972168, 6.67478047776068203590468141054, 7.12321883171349985433912907475, 8.523817816654045656960860796780, 9.049318663979790389679530577106, 10.34510737634187143050993707602, 11.15605986747601516646495582487