| L(s) = 1 | + 10·3-s + 7·7-s + 73·9-s − 40·11-s + 12·13-s + 58·17-s + 26·19-s + 70·21-s + 64·23-s + 460·27-s − 62·29-s + 252·31-s − 400·33-s − 26·37-s + 120·39-s + 6·41-s − 416·43-s + 396·47-s + 49·49-s + 580·51-s + 450·53-s + 260·57-s + 274·59-s − 576·61-s + 511·63-s + 476·67-s + 640·69-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 0.377·7-s + 2.70·9-s − 1.09·11-s + 0.256·13-s + 0.827·17-s + 0.313·19-s + 0.727·21-s + 0.580·23-s + 3.27·27-s − 0.397·29-s + 1.46·31-s − 2.11·33-s − 0.115·37-s + 0.492·39-s + 0.0228·41-s − 1.47·43-s + 1.22·47-s + 1/7·49-s + 1.59·51-s + 1.16·53-s + 0.604·57-s + 0.604·59-s − 1.20·61-s + 1.02·63-s + 0.867·67-s + 1.11·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.559223427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.559223427\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 - 26 T + p^{3} T^{2} \) |
| 23 | \( 1 - 64 T + p^{3} T^{2} \) |
| 29 | \( 1 + 62 T + p^{3} T^{2} \) |
| 31 | \( 1 - 252 T + p^{3} T^{2} \) |
| 37 | \( 1 + 26 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 416 T + p^{3} T^{2} \) |
| 47 | \( 1 - 396 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 274 T + p^{3} T^{2} \) |
| 61 | \( 1 + 576 T + p^{3} T^{2} \) |
| 67 | \( 1 - 476 T + p^{3} T^{2} \) |
| 71 | \( 1 + 448 T + p^{3} T^{2} \) |
| 73 | \( 1 - 158 T + p^{3} T^{2} \) |
| 79 | \( 1 + 936 T + p^{3} T^{2} \) |
| 83 | \( 1 + 530 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 214 T + p^{3} T^{2} \) |
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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
−9.969335601061136671829689108175, −9.027422903476006908606516737367, −8.275403662906193004689982041065, −7.74481946580882284081581322485, −6.92350390351497276644392806394, −5.35992662620088655008075151681, −4.27649952445758632079190040843, −3.21479019931065207065667189531, −2.48288358976787979768714778513, −1.26320442280693282091726182766,
1.26320442280693282091726182766, 2.48288358976787979768714778513, 3.21479019931065207065667189531, 4.27649952445758632079190040843, 5.35992662620088655008075151681, 6.92350390351497276644392806394, 7.74481946580882284081581322485, 8.275403662906193004689982041065, 9.027422903476006908606516737367, 9.969335601061136671829689108175
