| L(s) = 1 | + 2.23·3-s + 2.00·9-s + 2.23·11-s + 4·13-s − 3·17-s + 2.23·19-s + 8.94·23-s − 2.23·27-s − 4·29-s + 8.94·31-s + 5.00·33-s + 8·37-s + 8.94·39-s + 5·41-s − 8.94·43-s − 8.94·47-s − 7·49-s − 6.70·51-s + 4·53-s + 5.00·57-s − 8.94·59-s − 8·61-s + 6.70·67-s + 20.0·69-s + 8.94·71-s − 9·73-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 0.666·9-s + 0.674·11-s + 1.10·13-s − 0.727·17-s + 0.512·19-s + 1.86·23-s − 0.430·27-s − 0.742·29-s + 1.60·31-s + 0.870·33-s + 1.31·37-s + 1.43·39-s + 0.780·41-s − 1.36·43-s − 1.30·47-s − 49-s − 0.939·51-s + 0.549·53-s + 0.662·57-s − 1.16·59-s − 1.02·61-s + 0.819·67-s + 2.40·69-s + 1.06·71-s − 1.05·73-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.775392350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.775392350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 6.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.187017328247638614254152943637, −7.47936130293416306525307192240, −6.62940560131177360151873950474, −6.12895175587398555693508087575, −4.98182644668414629674938856630, −4.27492176168407701402275717824, −3.36129759813884846978934055298, −2.97582974023882825779574740362, −1.90725591508502306000371935443, −1.00602415346545784953591029377,
1.00602415346545784953591029377, 1.90725591508502306000371935443, 2.97582974023882825779574740362, 3.36129759813884846978934055298, 4.27492176168407701402275717824, 4.98182644668414629674938856630, 6.12895175587398555693508087575, 6.62940560131177360151873950474, 7.47936130293416306525307192240, 8.187017328247638614254152943637
