| L(s) = 1 | + 8·3-s + 10·5-s + 37·9-s + 40·11-s + 50·13-s + 80·15-s + 30·17-s + 40·19-s − 48·23-s − 25·25-s + 80·27-s − 34·29-s + 320·31-s + 320·33-s + 310·37-s + 400·39-s − 410·41-s − 152·43-s + 370·45-s − 416·47-s + 240·51-s − 410·53-s + 400·55-s + 320·57-s − 200·59-s − 30·61-s + 500·65-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.894·5-s + 1.37·9-s + 1.09·11-s + 1.06·13-s + 1.37·15-s + 0.428·17-s + 0.482·19-s − 0.435·23-s − 1/5·25-s + 0.570·27-s − 0.217·29-s + 1.85·31-s + 1.68·33-s + 1.37·37-s + 1.64·39-s − 1.56·41-s − 0.539·43-s + 1.22·45-s − 1.29·47-s + 0.658·51-s − 1.06·53-s + 0.980·55-s + 0.743·57-s − 0.441·59-s − 0.0629·61-s + 0.954·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.694224130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.694224130\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 - 320 T + p^{3} T^{2} \) |
| 37 | \( 1 - 310 T + p^{3} T^{2} \) |
| 41 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 416 T + p^{3} T^{2} \) |
| 53 | \( 1 + 410 T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 + 776 T + p^{3} T^{2} \) |
| 71 | \( 1 + 400 T + p^{3} T^{2} \) |
| 73 | \( 1 - 630 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1120 T + p^{3} T^{2} \) |
| 83 | \( 1 - 552 T + p^{3} T^{2} \) |
| 89 | \( 1 - 326 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−9.109502685393367301512617081027, −8.318936145496214388270854886313, −7.77930701340118556099443319817, −6.55416562543892942668278806154, −6.05959191111651630596785866394, −4.73174054014999539460157886702, −3.69538146126177927029854218436, −3.05955305407765584103010144143, −1.93245272526597172193731663685, −1.21931858290260886063232639427,
1.21931858290260886063232639427, 1.93245272526597172193731663685, 3.05955305407765584103010144143, 3.69538146126177927029854218436, 4.73174054014999539460157886702, 6.05959191111651630596785866394, 6.55416562543892942668278806154, 7.77930701340118556099443319817, 8.318936145496214388270854886313, 9.109502685393367301512617081027
