| L(s) = 1 | + 2-s − 3-s + 4-s + 3.27·5-s − 6-s + 3.11·7-s + 8-s + 9-s + 3.27·10-s + 5.01·11-s − 12-s − 2.25·13-s + 3.11·14-s − 3.27·15-s + 16-s + 1.75·17-s + 18-s − 7.20·19-s + 3.27·20-s − 3.11·21-s + 5.01·22-s + 4.20·23-s − 24-s + 5.70·25-s − 2.25·26-s − 27-s + 3.11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.46·5-s − 0.408·6-s + 1.17·7-s + 0.353·8-s + 0.333·9-s + 1.03·10-s + 1.51·11-s − 0.288·12-s − 0.625·13-s + 0.831·14-s − 0.844·15-s + 0.250·16-s + 0.426·17-s + 0.235·18-s − 1.65·19-s + 0.731·20-s − 0.678·21-s + 1.06·22-s + 0.876·23-s − 0.204·24-s + 1.14·25-s − 0.442·26-s − 0.192·27-s + 0.587·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.276202070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.276202070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 - 3.11T + 7T^{2} \) |
| 11 | \( 1 - 5.01T + 11T^{2} \) |
| 13 | \( 1 + 2.25T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 - 3.29T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 5.32T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 9.78T + 67T^{2} \) |
| 71 | \( 1 + 0.0747T + 71T^{2} \) |
| 73 | \( 1 - 8.00T + 73T^{2} \) |
| 79 | \( 1 - 7.81T + 79T^{2} \) |
| 83 | \( 1 - 6.06T + 83T^{2} \) |
| 89 | \( 1 - 7.54T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−9.624836062436942993833294543656, −9.029551392632154331570784471761, −7.84690466133859787170026949299, −6.77477270158378128365980188822, −6.23546086345566221609398882171, −5.35314467712422020606155213119, −4.76696701406111164561404178720, −3.73040232797359881566520272203, −2.09350130862748884879969094028, −1.53074346585097532370333751195,
1.53074346585097532370333751195, 2.09350130862748884879969094028, 3.73040232797359881566520272203, 4.76696701406111164561404178720, 5.35314467712422020606155213119, 6.23546086345566221609398882171, 6.77477270158378128365980188822, 7.84690466133859787170026949299, 9.029551392632154331570784471761, 9.624836062436942993833294543656
