| L(s) = 1 | + 0.0899·2-s − 1.99·4-s − 1.20·5-s + 3.43·7-s − 0.359·8-s − 0.108·10-s + 0.0198·11-s + 4.14·13-s + 0.309·14-s + 3.95·16-s + 4.43·17-s − 19-s + 2.39·20-s + 0.00178·22-s − 1.27·23-s − 3.55·25-s + 0.372·26-s − 6.84·28-s + 7.52·29-s − 3.98·31-s + 1.07·32-s + 0.398·34-s − 4.12·35-s + 1.29·37-s − 0.0899·38-s + 0.431·40-s + 9.40·41-s + ⋯ |
| L(s) = 1 | + 0.0636·2-s − 0.995·4-s − 0.537·5-s + 1.29·7-s − 0.126·8-s − 0.0341·10-s + 0.00597·11-s + 1.14·13-s + 0.0826·14-s + 0.987·16-s + 1.07·17-s − 0.229·19-s + 0.535·20-s + 0.000380·22-s − 0.266·23-s − 0.711·25-s + 0.0730·26-s − 1.29·28-s + 1.39·29-s − 0.716·31-s + 0.189·32-s + 0.0684·34-s − 0.697·35-s + 0.212·37-s − 0.0145·38-s + 0.0682·40-s + 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.952999480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.952999480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0899T + 2T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 - 0.0198T + 11T^{2} \) |
| 13 | \( 1 - 4.14T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 23 | \( 1 + 1.27T + 23T^{2} \) |
| 29 | \( 1 - 7.52T + 29T^{2} \) |
| 31 | \( 1 + 3.98T + 31T^{2} \) |
| 37 | \( 1 - 1.29T + 37T^{2} \) |
| 41 | \( 1 - 9.40T + 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + 8.51T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 7.20T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 5.10T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
| 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
−7.84421168201937236976196902508, −7.54215135541121195010819463868, −6.25401140730833113480807031883, −5.64158885017319842997635174173, −4.98527694332145590454952897719, −4.12939454045049362978423443489, −3.88465653593699726052455069581, −2.78351073667335713027421127354, −1.50869054568912692117415983610, −0.76510387309969725433842252533,
0.76510387309969725433842252533, 1.50869054568912692117415983610, 2.78351073667335713027421127354, 3.88465653593699726052455069581, 4.12939454045049362978423443489, 4.98527694332145590454952897719, 5.64158885017319842997635174173, 6.25401140730833113480807031883, 7.54215135541121195010819463868, 7.84421168201937236976196902508
