L(s) = 1 | + 2-s + 2.93·3-s + 4-s + 2.93·6-s + 7-s + 8-s + 5.63·9-s + 11-s + 2.93·12-s + 5.87·13-s + 14-s + 16-s + 4.63·17-s + 5.63·18-s − 5.57·19-s + 2.93·21-s + 22-s − 5.57·23-s + 2.93·24-s + 5.87·26-s + 7.75·27-s + 28-s − 9.45·29-s + 6·31-s + 32-s + 2.93·33-s + 4.63·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.69·3-s + 0.5·4-s + 1.19·6-s + 0.377·7-s + 0.353·8-s + 1.87·9-s + 0.301·11-s + 0.848·12-s + 1.63·13-s + 0.267·14-s + 0.250·16-s + 1.12·17-s + 1.32·18-s − 1.27·19-s + 0.641·21-s + 0.213·22-s − 1.16·23-s + 0.599·24-s + 1.15·26-s + 1.49·27-s + 0.188·28-s − 1.75·29-s + 1.07·31-s + 0.176·32-s + 0.511·33-s + 0.795·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.415840397\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.415840397\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 + 5.57T + 23T^{2} \) |
| 29 | \( 1 + 9.45T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 + 3.06T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 8.51T + 47T^{2} \) |
| 53 | \( 1 - 9.45T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 7.15T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 + 7.57T + 97T^{2} \) |
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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
−8.399371039992816700740834023329, −7.975215097131475003523277693711, −7.08637599500445698219490027450, −6.28514681121413260915923846383, −5.48221533111094514674996285465, −4.28783661000407982705507597787, −3.76147168989875605616220351554, −3.21286208119120149518493428897, −2.08314808476120317828220896332, −1.47912182013148331859850574946,
1.47912182013148331859850574946, 2.08314808476120317828220896332, 3.21286208119120149518493428897, 3.76147168989875605616220351554, 4.28783661000407982705507597787, 5.48221533111094514674996285465, 6.28514681121413260915923846383, 7.08637599500445698219490027450, 7.975215097131475003523277693711, 8.399371039992816700740834023329