| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 2·11-s − 4·13-s − 15-s − 2·17-s + 19-s − 2·21-s + 25-s − 27-s − 4·29-s − 8·31-s − 2·33-s + 2·35-s − 8·37-s + 4·39-s − 12·41-s + 6·43-s + 45-s − 3·49-s + 2·51-s − 2·53-s + 2·55-s − 57-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s + 0.229·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 1.43·31-s − 0.348·33-s + 0.338·35-s − 1.31·37-s + 0.640·39-s − 1.87·41-s + 0.914·43-s + 0.149·45-s − 3/7·49-s + 0.280·51-s − 0.274·53-s + 0.269·55-s − 0.132·57-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.87399844452518710393734282229, −7.10421117543815703028113632860, −6.63250465495224351733300772643, −5.54752958752285420937471472092, −5.16537546042188384535981105647, −4.35753761003575635954307485304, −3.42992123501035999199519809891, −2.16204729931807765491735279191, −1.49908553896607367048139509191, 0,
1.49908553896607367048139509191, 2.16204729931807765491735279191, 3.42992123501035999199519809891, 4.35753761003575635954307485304, 5.16537546042188384535981105647, 5.54752958752285420937471472092, 6.63250465495224351733300772643, 7.10421117543815703028113632860, 7.87399844452518710393734282229
