| L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·6-s + 4·7-s − 3·8-s + 9-s + 2·11-s − 2·12-s + 13-s + 4·14-s − 16-s − 2·17-s + 18-s − 6·19-s + 8·21-s + 2·22-s + 6·23-s − 6·24-s + 26-s − 4·27-s − 4·28-s + 2·29-s − 10·31-s + 5·32-s + 4·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 0.277·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.37·19-s + 1.74·21-s + 0.426·22-s + 1.25·23-s − 1.22·24-s + 0.196·26-s − 0.769·27-s − 0.755·28-s + 0.371·29-s − 1.79·31-s + 0.883·32-s + 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.407285254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.407285254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−11.65896684812187657882654865859, −10.88333347945116111183993338492, −9.406400646348943696684170504127, −8.620250295882527182273636480991, −8.208457502375816686030815529949, −6.77389438746521045568684633305, −5.32690719210735599071172022564, −4.40334075487539019542891709971, −3.42973993124646170162064326406, −1.95428342136872914409252140690,
1.95428342136872914409252140690, 3.42973993124646170162064326406, 4.40334075487539019542891709971, 5.32690719210735599071172022564, 6.77389438746521045568684633305, 8.208457502375816686030815529949, 8.620250295882527182273636480991, 9.406400646348943696684170504127, 10.88333347945116111183993338492, 11.65896684812187657882654865859
