| L(s) = 1 | + 1.22·5-s + (−1.48 − 2.19i)7-s − 3.31i·11-s − 1.60i·13-s − 6.11·17-s + 1.35i·19-s − 1.11i·23-s − 3.50·25-s + 9.39i·29-s − 7.00i·31-s + (−1.81 − 2.67i)35-s − 11.7·37-s + 5.86·41-s − 8.58·43-s + 3.15·47-s + ⋯ |
| L(s) = 1 | + 0.546·5-s + (−0.560 − 0.828i)7-s − 0.998i·11-s − 0.443i·13-s − 1.48·17-s + 0.310i·19-s − 0.232i·23-s − 0.701·25-s + 1.74i·29-s − 1.25i·31-s + (−0.306 − 0.452i)35-s − 1.92·37-s + 0.916·41-s − 1.30·43-s + 0.460·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8199461933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8199461933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.48 + 2.19i)T \) |
| good | 5 | \( 1 - 1.22T + 5T^{2} \) |
| 11 | \( 1 + 3.31iT - 11T^{2} \) |
| 13 | \( 1 + 1.60iT - 13T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 19 | \( 1 - 1.35iT - 19T^{2} \) |
| 23 | \( 1 + 1.11iT - 23T^{2} \) |
| 29 | \( 1 - 9.39iT - 29T^{2} \) |
| 31 | \( 1 + 7.00iT - 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 - 0.539iT - 53T^{2} \) |
| 59 | \( 1 - 6.87T + 59T^{2} \) |
| 61 | \( 1 + 7.40iT - 61T^{2} \) |
| 67 | \( 1 + 5.74T + 67T^{2} \) |
| 71 | \( 1 + 4.81iT - 71T^{2} \) |
| 73 | \( 1 + 14.5iT - 73T^{2} \) |
| 79 | \( 1 - 1.15T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 4.04iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.127251326229156229548608767806, −8.478247398230285954783596163922, −7.46819877388058006428857048784, −6.63594400133242190000227061232, −5.98815709762350091006818897607, −5.02898574253519449729656703925, −3.91318769960674800666833778727, −3.11067295873632857947081817261, −1.81994954285830389700883887876, −0.29615709143689738051140200712,
1.88633823272105763280161259962, 2.54320352844475234947584222484, 3.90348398572652593111620812341, 4.85479690697588282305692379025, 5.74280656366367604096795968260, 6.60486946508604752329428131013, 7.15671218322779841647937617004, 8.433782428650259301487694176705, 9.048712628892848594759107716190, 9.751449502750552464188587803435
