| L(s) = 1 | + 2-s + 4-s + 5-s + 3·7-s + 8-s + 10-s + 3·11-s − 5·13-s + 3·14-s + 16-s − 5·17-s + 19-s + 20-s + 3·22-s + 7·23-s + 25-s − 5·26-s + 3·28-s + 6·29-s + 6·31-s + 32-s − 5·34-s + 3·35-s − 37-s + 38-s + 40-s − 8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s − 1.38·13-s + 0.801·14-s + 1/4·16-s − 1.21·17-s + 0.229·19-s + 0.223·20-s + 0.639·22-s + 1.45·23-s + 1/5·25-s − 0.980·26-s + 0.566·28-s + 1.11·29-s + 1.07·31-s + 0.176·32-s − 0.857·34-s + 0.507·35-s − 0.164·37-s + 0.162·38-s + 0.158·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.852820239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.852820239\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.725198433302142604633219165929, −7.71741198857819973701576540404, −6.98647439274816025695495790580, −6.43522854306944829406826300238, −5.37095872123321980059523424349, −4.74713188295472581477697842393, −4.27214922286674094259065818861, −2.92577954721498397707506062505, −2.19167759582800545199991471174, −1.14324818618480286577693945927,
1.14324818618480286577693945927, 2.19167759582800545199991471174, 2.92577954721498397707506062505, 4.27214922286674094259065818861, 4.74713188295472581477697842393, 5.37095872123321980059523424349, 6.43522854306944829406826300238, 6.98647439274816025695495790580, 7.71741198857819973701576540404, 8.725198433302142604633219165929
