| L(s) = 1 | − 2·2-s + 4·4-s + 10·5-s − 7·7-s − 8·8-s − 20·10-s + 18·11-s + 2·13-s + 14·14-s + 16·16-s − 26·17-s − 19·19-s + 40·20-s − 36·22-s − 146·23-s − 25·25-s − 4·26-s − 28·28-s − 68·29-s + 340·31-s − 32·32-s + 52·34-s − 70·35-s − 22·37-s + 38·38-s − 80·40-s + 150·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s + 0.493·11-s + 0.0426·13-s + 0.267·14-s + 1/4·16-s − 0.370·17-s − 0.229·19-s + 0.447·20-s − 0.348·22-s − 1.32·23-s − 1/5·25-s − 0.0301·26-s − 0.188·28-s − 0.435·29-s + 1.96·31-s − 0.176·32-s + 0.262·34-s − 0.338·35-s − 0.0977·37-s + 0.162·38-s − 0.316·40-s + 0.571·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.639625829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.639625829\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 19 | \( 1 + p T \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 26 T + p^{3} T^{2} \) |
| 23 | \( 1 + 146 T + p^{3} T^{2} \) |
| 29 | \( 1 + 68 T + p^{3} T^{2} \) |
| 31 | \( 1 - 340 T + p^{3} T^{2} \) |
| 37 | \( 1 + 22 T + p^{3} T^{2} \) |
| 41 | \( 1 - 150 T + p^{3} T^{2} \) |
| 43 | \( 1 + 180 T + p^{3} T^{2} \) |
| 47 | \( 1 - 56 T + p^{3} T^{2} \) |
| 53 | \( 1 + 112 T + p^{3} T^{2} \) |
| 59 | \( 1 - 556 T + p^{3} T^{2} \) |
| 61 | \( 1 - 710 T + p^{3} T^{2} \) |
| 67 | \( 1 + 908 T + p^{3} T^{2} \) |
| 71 | \( 1 - 478 T + p^{3} T^{2} \) |
| 73 | \( 1 + 938 T + p^{3} T^{2} \) |
| 79 | \( 1 + 112 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1614 T + p^{3} T^{2} \) |
| 97 | \( 1 - 462 T + p^{3} T^{2} \) |
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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
−8.703415977237262884662532261229, −7.994159175067024885504386078802, −7.05051839354737265567869437402, −6.24602303511531386902186824912, −5.87325720845413666983928983028, −4.64689692333633095421623112434, −3.63986298044378630419392618522, −2.49445166239379032608020029128, −1.77621715043148095577343064212, −0.62770894826343944688211647811,
0.62770894826343944688211647811, 1.77621715043148095577343064212, 2.49445166239379032608020029128, 3.63986298044378630419392618522, 4.64689692333633095421623112434, 5.87325720845413666983928983028, 6.24602303511531386902186824912, 7.05051839354737265567869437402, 7.994159175067024885504386078802, 8.703415977237262884662532261229
