| L(s) = 1 | + 2.44·2-s − 3-s + 3.95·4-s − 2.44·6-s + 2.82·7-s + 4.78·8-s + 9-s − 1.69·11-s − 3.95·12-s + 1.94·13-s + 6.89·14-s + 3.76·16-s − 4.09·17-s + 2.44·18-s − 1.35·19-s − 2.82·21-s − 4.13·22-s + 2.48·23-s − 4.78·24-s + 4.75·26-s − 27-s + 11.1·28-s + 8.62·29-s + 0.625·31-s − 0.387·32-s + 1.69·33-s − 9.99·34-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.577·3-s + 1.97·4-s − 0.996·6-s + 1.06·7-s + 1.69·8-s + 0.333·9-s − 0.510·11-s − 1.14·12-s + 0.539·13-s + 1.84·14-s + 0.940·16-s − 0.993·17-s + 0.575·18-s − 0.311·19-s − 0.616·21-s − 0.881·22-s + 0.517·23-s − 0.976·24-s + 0.931·26-s − 0.192·27-s + 2.11·28-s + 1.60·29-s + 0.112·31-s − 0.0685·32-s + 0.294·33-s − 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.904027285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.904027285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 + 4.09T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 - 2.48T + 23T^{2} \) |
| 29 | \( 1 - 8.62T + 29T^{2} \) |
| 31 | \( 1 - 0.625T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 8.61T + 43T^{2} \) |
| 47 | \( 1 + 5.81T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 5.44T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 - 9.18T + 79T^{2} \) |
| 83 | \( 1 - 5.24T + 83T^{2} \) |
| 89 | \( 1 - 2.79T + 89T^{2} \) |
| 97 | \( 1 + 3.44T + 97T^{2} \) |
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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
−7.61399156299687851051611516200, −6.75089604977643481454805392888, −6.28562466610677087800762732604, −5.59462301855182006725319687644, −4.84697641549284650678226796121, −4.54617463853556817346407992838, −3.82498496971224604523881846701, −2.74857070458411026011924469624, −2.13413088690365712705368854933, −0.982823163461483582852716627934,
0.982823163461483582852716627934, 2.13413088690365712705368854933, 2.74857070458411026011924469624, 3.82498496971224604523881846701, 4.54617463853556817346407992838, 4.84697641549284650678226796121, 5.59462301855182006725319687644, 6.28562466610677087800762732604, 6.75089604977643481454805392888, 7.61399156299687851051611516200
