| L(s) = 1 | − 7·3-s + 5·5-s − 11·7-s + 22·9-s + 36·11-s + 65·13-s − 35·15-s − 87·17-s − 19·19-s + 77·21-s + 129·23-s + 25·25-s + 35·27-s + 231·29-s − 110·31-s − 252·33-s − 55·35-s − 142·37-s − 455·39-s − 330·41-s − 74·43-s + 110·45-s + 336·47-s − 222·49-s + 609·51-s + 501·53-s + 180·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.447·5-s − 0.593·7-s + 0.814·9-s + 0.986·11-s + 1.38·13-s − 0.602·15-s − 1.24·17-s − 0.229·19-s + 0.800·21-s + 1.16·23-s + 1/5·25-s + 0.249·27-s + 1.47·29-s − 0.637·31-s − 1.32·33-s − 0.265·35-s − 0.630·37-s − 1.86·39-s − 1.25·41-s − 0.262·43-s + 0.364·45-s + 1.04·47-s − 0.647·49-s + 1.67·51-s + 1.29·53-s + 0.441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.304225937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304225937\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 + p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 11 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 87 T + p^{3} T^{2} \) |
| 23 | \( 1 - 129 T + p^{3} T^{2} \) |
| 29 | \( 1 - 231 T + p^{3} T^{2} \) |
| 31 | \( 1 + 110 T + p^{3} T^{2} \) |
| 37 | \( 1 + 142 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 74 T + p^{3} T^{2} \) |
| 47 | \( 1 - 336 T + p^{3} T^{2} \) |
| 53 | \( 1 - 501 T + p^{3} T^{2} \) |
| 59 | \( 1 + 633 T + p^{3} T^{2} \) |
| 61 | \( 1 + 88 T + p^{3} T^{2} \) |
| 67 | \( 1 + 119 T + p^{3} T^{2} \) |
| 71 | \( 1 - 204 T + p^{3} T^{2} \) |
| 73 | \( 1 - 407 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1262 T + p^{3} T^{2} \) |
| 83 | \( 1 + 270 T + p^{3} T^{2} \) |
| 89 | \( 1 + 30 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1406 T + p^{3} T^{2} \) |
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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
−8.961842362238216103866438876690, −8.662526105407325284372899052862, −6.99276487516920826238867757707, −6.52103960350508767950651908075, −6.01928198920880997985280165683, −5.07316136752519275740773281784, −4.19187988342974052424306623226, −3.11052119757583059013990338338, −1.60100472706672842804334234493, −0.62741116925494716985051436472,
0.62741116925494716985051436472, 1.60100472706672842804334234493, 3.11052119757583059013990338338, 4.19187988342974052424306623226, 5.07316136752519275740773281784, 6.01928198920880997985280165683, 6.52103960350508767950651908075, 6.99276487516920826238867757707, 8.662526105407325284372899052862, 8.961842362238216103866438876690
