| L(s) = 1 | − 4·3-s − 5·5-s − 11·9-s + 36·11-s + 42·13-s + 20·15-s + 110·17-s + 116·19-s + 16·23-s + 25·25-s + 152·27-s + 198·29-s − 240·31-s − 144·33-s − 258·37-s − 168·39-s − 442·41-s − 292·43-s + 55·45-s − 392·47-s − 440·51-s + 142·53-s − 180·55-s − 464·57-s + 348·59-s + 570·61-s − 210·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.447·5-s − 0.407·9-s + 0.986·11-s + 0.896·13-s + 0.344·15-s + 1.56·17-s + 1.40·19-s + 0.145·23-s + 1/5·25-s + 1.08·27-s + 1.26·29-s − 1.39·31-s − 0.759·33-s − 1.14·37-s − 0.689·39-s − 1.68·41-s − 1.03·43-s + 0.182·45-s − 1.21·47-s − 1.20·51-s + 0.368·53-s − 0.441·55-s − 1.07·57-s + 0.767·59-s + 1.19·61-s − 0.400·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.665507398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665507398\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 110 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 16 T + p^{3} T^{2} \) |
| 29 | \( 1 - 198 T + p^{3} T^{2} \) |
| 31 | \( 1 + 240 T + p^{3} T^{2} \) |
| 37 | \( 1 + 258 T + p^{3} T^{2} \) |
| 41 | \( 1 + 442 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 392 T + p^{3} T^{2} \) |
| 53 | \( 1 - 142 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 570 T + p^{3} T^{2} \) |
| 67 | \( 1 - 692 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 134 T + p^{3} T^{2} \) |
| 79 | \( 1 - 784 T + p^{3} T^{2} \) |
| 83 | \( 1 + 564 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1034 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−8.677008027035987045813432930499, −8.165182801105838533052664435839, −7.06225859411952504838778602299, −6.50105804295316399635856378201, −5.47998847378264623883021332601, −5.07535682573438079351141878036, −3.64360051930141708120540245663, −3.26956951942213813275790513872, −1.49697776397294957042214310589, −0.67879977302095928445962358631,
0.67879977302095928445962358631, 1.49697776397294957042214310589, 3.26956951942213813275790513872, 3.64360051930141708120540245663, 5.07535682573438079351141878036, 5.47998847378264623883021332601, 6.50105804295316399635856378201, 7.06225859411952504838778602299, 8.165182801105838533052664435839, 8.677008027035987045813432930499
