| L(s) = 1 | + 1.51·2-s + 0.298·4-s − 2.07·5-s − 1.53·7-s − 2.57·8-s − 3.14·10-s − 1.76·11-s + 1.58·13-s − 2.33·14-s − 4.50·16-s − 5.51·17-s − 2.21·19-s − 0.620·20-s − 2.67·22-s + 4.68·23-s − 0.691·25-s + 2.40·26-s − 0.460·28-s + 10.7·29-s − 4.78·31-s − 1.67·32-s − 8.35·34-s + 3.19·35-s + 7.22·37-s − 3.36·38-s + 5.35·40-s + 6.55·41-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.149·4-s − 0.928·5-s − 0.582·7-s − 0.911·8-s − 0.995·10-s − 0.532·11-s + 0.439·13-s − 0.624·14-s − 1.12·16-s − 1.33·17-s − 0.508·19-s − 0.138·20-s − 0.570·22-s + 0.976·23-s − 0.138·25-s + 0.471·26-s − 0.0869·28-s + 1.99·29-s − 0.859·31-s − 0.296·32-s − 1.43·34-s + 0.540·35-s + 1.18·37-s − 0.545·38-s + 0.846·40-s + 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.561384746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561384746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 1.51T + 2T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 - 7.22T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 - 6.33T + 43T^{2} \) |
| 47 | \( 1 - 0.411T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 - 4.28T + 67T^{2} \) |
| 71 | \( 1 + 9.09T + 71T^{2} \) |
| 73 | \( 1 + 1.01T + 73T^{2} \) |
| 79 | \( 1 - 4.72T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 4.70T + 89T^{2} \) |
| 97 | \( 1 + 6.40T + 97T^{2} \) |
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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
−8.561203622874202091384460988691, −7.59634784613153681969330839382, −6.77252091795571824000076207145, −6.17671242190706785692480413969, −5.35639298743124441171746718075, −4.36128523977199566599354523827, −4.15307350125851787836377500031, −3.09830541477942530144289481013, −2.47952644956357435428704164796, −0.58173019660986386286632257653,
0.58173019660986386286632257653, 2.47952644956357435428704164796, 3.09830541477942530144289481013, 4.15307350125851787836377500031, 4.36128523977199566599354523827, 5.35639298743124441171746718075, 6.17671242190706785692480413969, 6.77252091795571824000076207145, 7.59634784613153681969330839382, 8.561203622874202091384460988691
