| L(s) = 1 | + 2.46·2-s + 4.08·4-s − 5-s + 7-s + 5.13·8-s − 2.46·10-s + 5.54·11-s + 1.38·13-s + 2.46·14-s + 4.49·16-s − 5.74·17-s + 2.20·19-s − 4.08·20-s + 13.6·22-s − 5.74·23-s + 25-s + 3.41·26-s + 4.08·28-s + 6.16·29-s + 1.18·31-s + 0.820·32-s − 14.1·34-s − 35-s − 2.93·37-s + 5.42·38-s − 5.13·40-s + 2.58·41-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 2.04·4-s − 0.447·5-s + 0.377·7-s + 1.81·8-s − 0.779·10-s + 1.67·11-s + 0.384·13-s + 0.659·14-s + 1.12·16-s − 1.39·17-s + 0.504·19-s − 0.912·20-s + 2.91·22-s − 1.19·23-s + 0.200·25-s + 0.669·26-s + 0.771·28-s + 1.14·29-s + 0.212·31-s + 0.145·32-s − 2.43·34-s − 0.169·35-s − 0.482·37-s + 0.880·38-s − 0.811·40-s + 0.403·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.433076212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.433076212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 + 5.74T + 23T^{2} \) |
| 29 | \( 1 - 6.16T + 29T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + 2.93T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 4.18T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 4.93T + 59T^{2} \) |
| 61 | \( 1 - 7.74T + 61T^{2} \) |
| 67 | \( 1 + 6.91T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 3.38T + 73T^{2} \) |
| 79 | \( 1 + 2.11T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−10.39097245238084751361709306202, −9.134249047446782934100387179918, −8.258594581821128544961432751292, −7.01788168277708826106803488149, −6.49504209158974204972098514929, −5.62290696045913103994483902539, −4.36173573418078513277175123221, −4.13712067953685228773405927587, −2.96949962833681951052352765362, −1.64982597012910951397424024223,
1.64982597012910951397424024223, 2.96949962833681951052352765362, 4.13712067953685228773405927587, 4.36173573418078513277175123221, 5.62290696045913103994483902539, 6.49504209158974204972098514929, 7.01788168277708826106803488149, 8.258594581821128544961432751292, 9.134249047446782934100387179918, 10.39097245238084751361709306202
