| L(s) = 1 | − 2-s + 4-s − 1.47·7-s − 8-s − 2.29·11-s − 5.29·13-s + 1.47·14-s + 16-s − 1.47·17-s − 19-s + 2.29·22-s + 1.86·23-s + 5.29·26-s − 1.47·28-s + 7.16·29-s + 0.0470·31-s − 32-s + 1.47·34-s + 6.59·37-s + 38-s + 0.179·41-s + 7.98·43-s − 2.29·44-s − 1.86·46-s − 12.4·47-s − 4.82·49-s − 5.29·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.558·7-s − 0.353·8-s − 0.692·11-s − 1.46·13-s + 0.394·14-s + 0.250·16-s − 0.358·17-s − 0.229·19-s + 0.489·22-s + 0.389·23-s + 1.03·26-s − 0.279·28-s + 1.33·29-s + 0.00845·31-s − 0.176·32-s + 0.253·34-s + 1.08·37-s + 0.162·38-s + 0.0280·41-s + 1.21·43-s − 0.346·44-s − 0.275·46-s − 1.81·47-s − 0.688·49-s − 0.734·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6669241406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6669241406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 1.47T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 - 7.16T + 29T^{2} \) |
| 31 | \( 1 - 0.0470T + 31T^{2} \) |
| 37 | \( 1 - 6.59T + 37T^{2} \) |
| 41 | \( 1 - 0.179T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 6.34T + 59T^{2} \) |
| 61 | \( 1 + 9.93T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 + 6.46T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
| 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
−7.74014146294453995791405619238, −7.25423533433381964709728552992, −6.49429958039934027623896745067, −5.92787427100435568610545633445, −4.89749380861827777620818265916, −4.44488765662644566427812513868, −3.04816280047491355333267091879, −2.73813691271432471896261109695, −1.71640144263303283861071150557, −0.42748495198840934577381925340,
0.42748495198840934577381925340, 1.71640144263303283861071150557, 2.73813691271432471896261109695, 3.04816280047491355333267091879, 4.44488765662644566427812513868, 4.89749380861827777620818265916, 5.92787427100435568610545633445, 6.49429958039934027623896745067, 7.25423533433381964709728552992, 7.74014146294453995791405619238
