| L(s) = 1 | + 2.78·2-s − 3-s + 5.76·4-s − 2.78·6-s − 3.15·7-s + 10.5·8-s + 9-s + 2.94·11-s − 5.76·12-s + 0.188·13-s − 8.77·14-s + 17.7·16-s + 2.62·17-s + 2.78·18-s + 1.94·19-s + 3.15·21-s + 8.21·22-s − 1.30·23-s − 10.5·24-s + 0.524·26-s − 27-s − 18.1·28-s + 1.23·29-s + 2.65·31-s + 28.4·32-s − 2.94·33-s + 7.32·34-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.577·3-s + 2.88·4-s − 1.13·6-s − 1.19·7-s + 3.71·8-s + 0.333·9-s + 0.888·11-s − 1.66·12-s + 0.0521·13-s − 2.34·14-s + 4.43·16-s + 0.637·17-s + 0.656·18-s + 0.445·19-s + 0.687·21-s + 1.75·22-s − 0.272·23-s − 2.14·24-s + 0.102·26-s − 0.192·27-s − 3.43·28-s + 0.228·29-s + 0.476·31-s + 5.02·32-s − 0.512·33-s + 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.243358241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.243358241\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 - 0.188T + 13T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 - 2.65T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 0.369T + 53T^{2} \) |
| 59 | \( 1 + 6.90T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 - 6.69T + 71T^{2} \) |
| 73 | \( 1 + 9.46T + 73T^{2} \) |
| 79 | \( 1 - 0.420T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.659047591128524682382896365787, −8.066111595572831950671874749193, −7.10912284184510106956275004958, −6.43844749564416999908901990845, −6.04602195550296837053613818311, −5.16126525731879268374594963396, −4.28768954882655308870248742185, −3.51904496805432168623135122944, −2.77986636199752314665114051051, −1.36321171176972805673525430710,
1.36321171176972805673525430710, 2.77986636199752314665114051051, 3.51904496805432168623135122944, 4.28768954882655308870248742185, 5.16126525731879268374594963396, 6.04602195550296837053613818311, 6.43844749564416999908901990845, 7.10912284184510106956275004958, 8.066111595572831950671874749193, 9.659047591128524682382896365787
