L(s) = 1 | − 3.04i·5-s + 5.27i·7-s + 3·9-s − 6.50·11-s − 7.27·17-s − 4.35·19-s − 4i·23-s − 4.27·25-s + 16.0·35-s − 5.67·43-s − 9.13i·45-s − 2.72i·47-s − 20.8·49-s + 19.8i·55-s + 10.8i·61-s + ⋯ |
L(s) = 1 | − 1.36i·5-s + 1.99i·7-s + 9-s − 1.96·11-s − 1.76·17-s − 1.00·19-s − 0.834i·23-s − 0.854·25-s + 2.71·35-s − 0.865·43-s − 1.36i·45-s − 0.397i·47-s − 2.97·49-s + 2.67i·55-s + 1.38i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1335757901\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1335757901\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35T \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 3.04iT - 5T^{2} \) |
| 7 | \( 1 - 5.27iT - 7T^{2} \) |
| 11 | \( 1 + 6.50T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 + 2.72iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 10.8iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.01301001947091259370450403901, −8.995410528875449701325478033096, −8.627406756372271797291653003942, −7.991801306603278930125788226282, −6.69348735058433647175147127318, −5.71499435825021164142733899242, −4.96255007415563520042436330131, −4.44679112211103057178895879795, −2.61257758447966562836728982884, −1.95791676164844489658401001448,
0.05137820674468295480397290008, 1.96166224219467299219958504918, 3.12142272396226625215652180262, 4.11936313809395067975247173826, 4.82911573682017179764690877900, 6.35068932084957571124918586841, 7.05136585208376924090542535745, 7.43887073941199374416263472190, 8.268078268454776369000080264533, 9.804577339978728741057540656998