| L(s) = 1 | + 3-s + 4·7-s + 9-s + 11-s − 6·13-s + 17-s − 6·19-s + 4·21-s − 5·25-s + 27-s + 2·29-s + 8·31-s + 33-s − 2·37-s − 6·39-s − 2·41-s − 2·43-s − 6·47-s + 9·49-s + 51-s + 2·53-s − 6·57-s − 2·59-s − 8·61-s + 4·63-s + 8·67-s − 4·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 0.242·17-s − 1.37·19-s + 0.872·21-s − 25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.174·33-s − 0.328·37-s − 0.960·39-s − 0.312·41-s − 0.304·43-s − 0.875·47-s + 9/7·49-s + 0.140·51-s + 0.274·53-s − 0.794·57-s − 0.260·59-s − 1.02·61-s + 0.503·63-s + 0.977·67-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.03600855774345, −14.66407894356469, −14.24181284318533, −13.75914026597072, −13.19315438762938, −12.42456774639439, −11.99378546927266, −11.68101124168575, −10.83695728380466, −10.48228607640819, −9.703918851722199, −9.461952475851330, −8.476359970831475, −8.206550578370857, −7.827460269370602, −7.066654280358199, −6.611188427400323, −5.752663664476307, −5.011196689067806, −4.582163074858596, −4.138530202639184, −3.211503229637455, −2.358659711352755, −2.005374391872328, −1.199126177618042, 0,
1.199126177618042, 2.005374391872328, 2.358659711352755, 3.211503229637455, 4.138530202639184, 4.582163074858596, 5.011196689067806, 5.752663664476307, 6.611188427400323, 7.066654280358199, 7.827460269370602, 8.206550578370857, 8.476359970831475, 9.461952475851330, 9.703918851722199, 10.48228607640819, 10.83695728380466, 11.68101124168575, 11.99378546927266, 12.42456774639439, 13.19315438762938, 13.75914026597072, 14.24181284318533, 14.66407894356469, 15.03600855774345
