| L(s) = 1 | − 3-s + 2·5-s + 9-s + 2·11-s + 2·13-s − 2·15-s + 6·17-s − 19-s + 2·23-s − 25-s − 27-s + 4·29-s − 8·31-s − 2·33-s − 2·37-s − 2·39-s − 8·41-s − 8·43-s + 2·45-s + 2·47-s − 7·49-s − 6·51-s − 4·53-s + 4·55-s + 57-s + 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.516·15-s + 1.45·17-s − 0.229·19-s + 0.417·23-s − 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.43·31-s − 0.348·33-s − 0.328·37-s − 0.320·39-s − 1.24·41-s − 1.21·43-s + 0.298·45-s + 0.291·47-s − 49-s − 0.840·51-s − 0.549·53-s + 0.539·55-s + 0.132·57-s + 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.238193981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238193981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.19970228386202906162935861460, −11.27298065234336220825066097717, −10.23756701541404993955921048072, −9.519824281945706153386589648338, −8.336263881925856474820476225443, −6.95937341559717252012161118029, −6.00528925290387781663305320049, −5.11512058941944572023463227823, −3.51636850061813590584504045751, −1.55947103904026918949790889898,
1.55947103904026918949790889898, 3.51636850061813590584504045751, 5.11512058941944572023463227823, 6.00528925290387781663305320049, 6.95937341559717252012161118029, 8.336263881925856474820476225443, 9.519824281945706153386589648338, 10.23756701541404993955921048072, 11.27298065234336220825066097717, 12.19970228386202906162935861460
