| L(s) = 1 | − 0.381·3-s − 3·7-s − 2.85·9-s − 4.23·11-s − 13-s − 1.14·17-s − 5.85·19-s + 1.14·21-s − 1.76·23-s + 2.23·27-s − 9.47·29-s + 0.236·31-s + 1.61·33-s + 8.32·37-s + 0.381·39-s + 1.47·41-s − 6.23·43-s − 11.9·47-s + 2·49-s + 0.437·51-s − 10.4·53-s + 2.23·57-s + 4.47·59-s − 8.85·61-s + 8.56·63-s − 10.2·67-s + 0.673·69-s + ⋯ |
| L(s) = 1 | − 0.220·3-s − 1.13·7-s − 0.951·9-s − 1.27·11-s − 0.277·13-s − 0.277·17-s − 1.34·19-s + 0.250·21-s − 0.367·23-s + 0.430·27-s − 1.75·29-s + 0.0423·31-s + 0.281·33-s + 1.36·37-s + 0.0611·39-s + 0.229·41-s − 0.950·43-s − 1.74·47-s + 0.285·49-s + 0.0612·51-s − 1.43·53-s + 0.296·57-s + 0.582·59-s − 1.13·61-s + 1.07·63-s − 1.25·67-s + 0.0811·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 + 9.47T + 29T^{2} \) |
| 31 | \( 1 - 0.236T + 31T^{2} \) |
| 37 | \( 1 - 8.32T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 9.56T + 97T^{2} \) |
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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
−6.81412464046739518895463628622, −6.25271524060422592151794197147, −5.72689938942701716284732305747, −4.99445495499278407078501359120, −4.19605314906914832297406179813, −3.22995603812463205763638987115, −2.72263468534469737280643966256, −1.86330270031844821497343188054, 0, 0,
1.86330270031844821497343188054, 2.72263468534469737280643966256, 3.22995603812463205763638987115, 4.19605314906914832297406179813, 4.99445495499278407078501359120, 5.72689938942701716284732305747, 6.25271524060422592151794197147, 6.81412464046739518895463628622
