| L(s) = 1 | − 0.381·2-s − 2.23·3-s − 1.85·4-s + 0.854·6-s − 0.236·7-s + 1.47·8-s + 2.00·9-s − 3.47·11-s + 4.14·12-s + 2·13-s + 0.0901·14-s + 3.14·16-s − 1.85·17-s − 0.763·18-s + 0.527·21-s + 1.32·22-s − 3.23·23-s − 3.29·24-s − 0.763·26-s + 2.23·27-s + 0.437·28-s + 6·29-s + 4.85·31-s − 4.14·32-s + 7.76·33-s + 0.708·34-s − 3.70·36-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 1.29·3-s − 0.927·4-s + 0.348·6-s − 0.0892·7-s + 0.520·8-s + 0.666·9-s − 1.04·11-s + 1.19·12-s + 0.554·13-s + 0.0240·14-s + 0.786·16-s − 0.449·17-s − 0.180·18-s + 0.115·21-s + 0.282·22-s − 0.674·23-s − 0.671·24-s − 0.149·26-s + 0.430·27-s + 0.0827·28-s + 1.11·29-s + 0.871·31-s − 0.732·32-s + 1.35·33-s + 0.121·34-s − 0.618·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3892739156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3892739156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 8.61T + 41T^{2} \) |
| 43 | \( 1 - 9.56T + 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 - 8.61T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 9.70T + 89T^{2} \) |
| 97 | \( 1 + 9.61T + 97T^{2} \) |
| 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
−7.77857227967649788125774006460, −6.97215607957067174994238822349, −6.24280760099609814104711435923, −5.60396282023249303941941109653, −5.03671459334692084818376305645, −4.48110911184354852977182520115, −3.64232794652694099581733285167, −2.60110369019768769603385459769, −1.33293718011767349634791484893, −0.37098842880183033455771216198,
0.37098842880183033455771216198, 1.33293718011767349634791484893, 2.60110369019768769603385459769, 3.64232794652694099581733285167, 4.48110911184354852977182520115, 5.03671459334692084818376305645, 5.60396282023249303941941109653, 6.24280760099609814104711435923, 6.97215607957067174994238822349, 7.77857227967649788125774006460
