| L(s) = 1 | − 3-s + 4.22·5-s + 4.94·7-s + 9-s − 11-s + 4.22·13-s − 4.22·15-s + 3.28·17-s − 1.28·19-s − 4.94·21-s + 2.22·23-s + 12.8·25-s − 27-s − 3.28·29-s + 2.56·31-s + 33-s + 20.9·35-s − 0.568·37-s − 4.22·39-s − 5.17·41-s − 11.1·43-s + 4.22·45-s − 10.2·47-s + 17.4·49-s − 3.28·51-s − 10.1·53-s − 4.22·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.89·5-s + 1.86·7-s + 0.333·9-s − 0.301·11-s + 1.17·13-s − 1.09·15-s + 0.796·17-s − 0.294·19-s − 1.07·21-s + 0.464·23-s + 2.57·25-s − 0.192·27-s − 0.609·29-s + 0.461·31-s + 0.174·33-s + 3.53·35-s − 0.0934·37-s − 0.677·39-s − 0.808·41-s − 1.70·43-s + 0.630·45-s − 1.49·47-s + 2.49·49-s − 0.459·51-s − 1.39·53-s − 0.570·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.807289705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.807289705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 4.22T + 5T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 13 | \( 1 - 4.22T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 + 0.568T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 + 5.66T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 5.77T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 0.486T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 6.45T + 97T^{2} \) |
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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
−9.061865534968382113295095590324, −8.418051244973322893771053279776, −7.58371966863677261905192634218, −6.46270753734872912180987069410, −5.88502911591080250143968431435, −5.13098507705786102524346110941, −4.67714957076373680391149069941, −3.11381351469914008116837338479, −1.65302638908903717800995762351, −1.49259085501182170606644098503,
1.49259085501182170606644098503, 1.65302638908903717800995762351, 3.11381351469914008116837338479, 4.67714957076373680391149069941, 5.13098507705786102524346110941, 5.88502911591080250143968431435, 6.46270753734872912180987069410, 7.58371966863677261905192634218, 8.418051244973322893771053279776, 9.061865534968382113295095590324
