| L(s) = 1 | + 1.36·2-s + 3.37·3-s − 0.150·4-s − 5-s + 4.59·6-s − 7-s − 2.92·8-s + 8.40·9-s − 1.36·10-s + 6.45·11-s − 0.507·12-s + 3.95·13-s − 1.36·14-s − 3.37·15-s − 3.67·16-s − 4.58·17-s + 11.4·18-s + 6.23·19-s + 0.150·20-s − 3.37·21-s + 8.78·22-s + 1.03·23-s − 9.87·24-s + 25-s + 5.38·26-s + 18.2·27-s + 0.150·28-s + ⋯ |
| L(s) = 1 | + 0.961·2-s + 1.95·3-s − 0.0751·4-s − 0.447·5-s + 1.87·6-s − 0.377·7-s − 1.03·8-s + 2.80·9-s − 0.430·10-s + 1.94·11-s − 0.146·12-s + 1.09·13-s − 0.363·14-s − 0.872·15-s − 0.919·16-s − 1.11·17-s + 2.69·18-s + 1.43·19-s + 0.0336·20-s − 0.737·21-s + 1.87·22-s + 0.214·23-s − 2.01·24-s + 0.200·25-s + 1.05·26-s + 3.51·27-s + 0.0284·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.813370279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.813370279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 3 | \( 1 - 3.37T + 3T^{2} \) |
| 11 | \( 1 - 6.45T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 - 1.03T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 0.589T + 37T^{2} \) |
| 41 | \( 1 - 3.44T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 - 5.95T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 - 7.71T + 67T^{2} \) |
| 71 | \( 1 + 3.30T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.83262976191784587340696308365, −7.12308612295072452883106692886, −6.58615024509114287416079850619, −5.72694298593823120969197331081, −4.57930223867553827264789400004, −3.92545305015360127398287069581, −3.65425875326140931674688613249, −3.11946783626005471255437994965, −2.05148852248572172026793624767, −1.10741807552992769085553325810,
1.10741807552992769085553325810, 2.05148852248572172026793624767, 3.11946783626005471255437994965, 3.65425875326140931674688613249, 3.92545305015360127398287069581, 4.57930223867553827264789400004, 5.72694298593823120969197331081, 6.58615024509114287416079850619, 7.12308612295072452883106692886, 7.83262976191784587340696308365
