L(s) = 1 | − 2.80·2-s − 3-s + 5.85·4-s + 2.80·6-s − 4.53·7-s − 10.8·8-s + 9-s − 5.85·12-s + 0.661·13-s + 12.7·14-s + 18.5·16-s + 3.87·17-s − 2.80·18-s − 1.73·19-s + 4.53·21-s − 1.14·23-s + 10.8·24-s − 1.85·26-s − 27-s − 26.5·28-s − 7.33·29-s − 6.85·31-s − 30.4·32-s − 10.8·34-s + 5.85·36-s − 5.70·37-s + 4.85·38-s + ⋯ |
L(s) = 1 | − 1.98·2-s − 0.577·3-s + 2.92·4-s + 1.14·6-s − 1.71·7-s − 3.81·8-s + 0.333·9-s − 1.68·12-s + 0.183·13-s + 3.39·14-s + 4.64·16-s + 0.939·17-s − 0.660·18-s − 0.397·19-s + 0.989·21-s − 0.238·23-s + 2.20·24-s − 0.363·26-s − 0.192·27-s − 5.01·28-s − 1.36·29-s − 1.23·31-s − 5.37·32-s − 1.86·34-s + 0.975·36-s − 0.938·37-s + 0.787·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1962330000\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1962330000\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 7 | \( 1 + 4.53T + 7T^{2} \) |
| 13 | \( 1 - 0.661T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 + 7.33T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 6.26T + 73T^{2} \) |
| 79 | \( 1 + 8.25T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 1.70T + 97T^{2} \) |
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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
−7.52357589383124240774335952860, −7.31689782155146706245225209509, −6.58376437011212909828632241208, −5.89553616352695236559432174662, −5.60628129346799067161738047876, −3.82126582039088715976273628259, −3.22861902182104489447222680207, −2.32391819990938888282884581246, −1.36812480520502613792195314480, −0.30956903956155356015770941006,
0.30956903956155356015770941006, 1.36812480520502613792195314480, 2.32391819990938888282884581246, 3.22861902182104489447222680207, 3.82126582039088715976273628259, 5.60628129346799067161738047876, 5.89553616352695236559432174662, 6.58376437011212909828632241208, 7.31689782155146706245225209509, 7.52357589383124240774335952860