| L(s) = 1 | − 3-s − 2·9-s − 5·11-s + 5·13-s − 17-s − 6·19-s − 4·23-s − 5·25-s + 5·27-s − 4·29-s + 5·33-s − 8·37-s − 5·39-s + 4·41-s − 6·43-s + 6·47-s + 51-s − 11·53-s + 6·57-s + 10·59-s + 10·67-s + 4·69-s − 9·71-s + 4·73-s + 5·75-s − 9·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 1.50·11-s + 1.38·13-s − 0.242·17-s − 1.37·19-s − 0.834·23-s − 25-s + 0.962·27-s − 0.742·29-s + 0.870·33-s − 1.31·37-s − 0.800·39-s + 0.624·41-s − 0.914·43-s + 0.875·47-s + 0.140·51-s − 1.51·53-s + 0.794·57-s + 1.30·59-s + 1.22·67-s + 0.481·69-s − 1.06·71-s + 0.468·73-s + 0.577·75-s − 1.01·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.05717328147490, −14.42276262649320, −13.71916759037734, −13.56394695939214, −12.78136711795994, −12.57194176501754, −11.80205861140399, −11.18391057639351, −11.01639998247252, −10.39232342856532, −10.03427912980581, −9.192257451737906, −8.556860173216121, −8.237254550473836, −7.780419518967847, −6.909452695721244, −6.401317976463480, −5.758701240317544, −5.547650915874713, −4.838262658949160, −4.039869369399232, −3.604116378661595, −2.698544686448879, −2.163681771070643, −1.355447021893490, 0, 0,
1.355447021893490, 2.163681771070643, 2.698544686448879, 3.604116378661595, 4.039869369399232, 4.838262658949160, 5.547650915874713, 5.758701240317544, 6.401317976463480, 6.909452695721244, 7.780419518967847, 8.237254550473836, 8.556860173216121, 9.192257451737906, 10.03427912980581, 10.39232342856532, 11.01639998247252, 11.18391057639351, 11.80205861140399, 12.57194176501754, 12.78136711795994, 13.56394695939214, 13.71916759037734, 14.42276262649320, 15.05717328147490
