| L(s) = 1 | + 5-s + 2.37·7-s + 4.37·11-s + 13-s + 1.62·17-s + 4·19-s − 1.62·23-s + 25-s + 8.74·29-s − 4.74·31-s + 2.37·35-s + 9.11·37-s + 4.37·41-s + 0.744·43-s − 11.4·47-s − 1.37·49-s + 1.62·53-s + 4.37·55-s + 2.74·59-s + 12.3·61-s + 65-s + 4·67-s + 4.37·71-s − 15.4·73-s + 10.3·77-s − 11.8·79-s − 14.7·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.896·7-s + 1.31·11-s + 0.277·13-s + 0.394·17-s + 0.917·19-s − 0.339·23-s + 0.200·25-s + 1.62·29-s − 0.852·31-s + 0.400·35-s + 1.49·37-s + 0.682·41-s + 0.113·43-s − 1.67·47-s − 0.196·49-s + 0.223·53-s + 0.589·55-s + 0.357·59-s + 1.58·61-s + 0.124·65-s + 0.488·67-s + 0.518·71-s − 1.81·73-s + 1.18·77-s − 1.33·79-s − 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.335441816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.335441816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 - 9.11T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 - 0.744T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 1.62T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.72927398809971664588979105047, −7.00591897768724380907370697029, −6.30238160308665860766059054056, −5.70908495849140922418411518502, −4.91653947876554260280886616552, −4.27228604596570394757109713647, −3.48185027999375245305525985248, −2.57108207479084605908880228301, −1.54898459005358564087585722476, −0.994979197858359544550419001509,
0.994979197858359544550419001509, 1.54898459005358564087585722476, 2.57108207479084605908880228301, 3.48185027999375245305525985248, 4.27228604596570394757109713647, 4.91653947876554260280886616552, 5.70908495849140922418411518502, 6.30238160308665860766059054056, 7.00591897768724380907370697029, 7.72927398809971664588979105047
