| L(s) = 1 | + 2.37·5-s + 1.37·7-s + 4·11-s + 6.74·13-s + 4.74·17-s − 2.37·19-s − 6.37·23-s + 0.627·25-s + 1.62·29-s − 2.62·31-s + 3.25·35-s − 7.37·37-s − 8.74·41-s − 0.627·43-s + 10.3·47-s − 5.11·49-s + 5.62·53-s + 9.48·55-s + 4·59-s + 5.37·61-s + 16·65-s − 9.74·67-s − 11.1·71-s + 73-s + 5.48·77-s − 13.4·79-s + 4.74·83-s + ⋯ |
| L(s) = 1 | + 1.06·5-s + 0.518·7-s + 1.20·11-s + 1.87·13-s + 1.15·17-s − 0.544·19-s − 1.32·23-s + 0.125·25-s + 0.302·29-s − 0.471·31-s + 0.550·35-s − 1.21·37-s − 1.36·41-s − 0.0957·43-s + 1.51·47-s − 0.730·49-s + 0.773·53-s + 1.27·55-s + 0.520·59-s + 0.687·61-s + 1.98·65-s − 1.19·67-s − 1.31·71-s + 0.117·73-s + 0.625·77-s − 1.51·79-s + 0.520·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.670446998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.670446998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 7.37T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 + 0.627T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 5.37T + 61T^{2} \) |
| 67 | \( 1 + 9.74T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 4.74T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−8.965574822353200633545742364924, −8.675375587673470690914564572181, −7.69320486902578840560546451838, −6.57881767532742096768827999194, −6.02361684374996984404487063074, −5.39270061984728222755958573130, −4.11577704856135377267977535941, −3.45351277432856482542374866957, −1.91953191493991169075284139653, −1.29077936769382084674269749391,
1.29077936769382084674269749391, 1.91953191493991169075284139653, 3.45351277432856482542374866957, 4.11577704856135377267977535941, 5.39270061984728222755958573130, 6.02361684374996984404487063074, 6.57881767532742096768827999194, 7.69320486902578840560546451838, 8.675375587673470690914564572181, 8.965574822353200633545742364924
