| L(s) = 1 | + 1.70·3-s + 4.34·5-s − 0.630·7-s − 0.0783·9-s − 5.70·11-s + 3.07·13-s + 7.41·15-s + 17-s + 3.41·19-s − 1.07·21-s + 5.89·23-s + 13.8·25-s − 5.26·27-s − 0.340·29-s + 6.78·31-s − 9.75·33-s − 2.73·35-s − 0.340·37-s + 5.26·39-s + 4.15·41-s + 5.26·43-s − 0.340·45-s − 8.68·47-s − 6.60·49-s + 1.70·51-s − 0.156·53-s − 24.7·55-s + ⋯ |
| L(s) = 1 | + 0.986·3-s + 1.94·5-s − 0.238·7-s − 0.0261·9-s − 1.72·11-s + 0.853·13-s + 1.91·15-s + 0.242·17-s + 0.784·19-s − 0.235·21-s + 1.22·23-s + 2.76·25-s − 1.01·27-s − 0.0631·29-s + 1.21·31-s − 1.69·33-s − 0.462·35-s − 0.0559·37-s + 0.842·39-s + 0.649·41-s + 0.802·43-s − 0.0507·45-s − 1.26·47-s − 0.943·49-s + 0.239·51-s − 0.0215·53-s − 3.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.910596435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.910596435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 5 | \( 1 - 4.34T + 5T^{2} \) |
| 7 | \( 1 + 0.630T + 7T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 5.89T + 23T^{2} \) |
| 29 | \( 1 + 0.340T + 29T^{2} \) |
| 31 | \( 1 - 6.78T + 31T^{2} \) |
| 37 | \( 1 + 0.340T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 + 0.156T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 7.17T + 61T^{2} \) |
| 67 | \( 1 + 6.83T + 67T^{2} \) |
| 71 | \( 1 - 3.95T + 71T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−9.718718558788766741783802630185, −9.158038200711875178484835363013, −8.371294837606900926222970000708, −7.52576174509093085198777834973, −6.33973306135595392677656508122, −5.64234466104544791664922707725, −4.87810818905486230839380964551, −3.01374849425865214508339349717, −2.74217130219172418228091475090, −1.47250098250406969626501673828,
1.47250098250406969626501673828, 2.74217130219172418228091475090, 3.01374849425865214508339349717, 4.87810818905486230839380964551, 5.64234466104544791664922707725, 6.33973306135595392677656508122, 7.52576174509093085198777834973, 8.371294837606900926222970000708, 9.158038200711875178484835363013, 9.718718558788766741783802630185
