| L(s) = 1 | − 3·3-s + 12·5-s + 9·9-s + 60·11-s + 44·13-s − 36·15-s − 128·17-s − 52·19-s + 160·23-s + 19·25-s − 27·27-s − 230·29-s − 136·31-s − 180·33-s − 318·37-s − 132·39-s − 192·41-s − 220·43-s + 108·45-s − 184·47-s + 384·51-s − 498·53-s + 720·55-s + 156·57-s + 492·59-s + 20·61-s + 528·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.07·5-s + 1/3·9-s + 1.64·11-s + 0.938·13-s − 0.619·15-s − 1.82·17-s − 0.627·19-s + 1.45·23-s + 0.151·25-s − 0.192·27-s − 1.47·29-s − 0.787·31-s − 0.949·33-s − 1.41·37-s − 0.541·39-s − 0.731·41-s − 0.780·43-s + 0.357·45-s − 0.571·47-s + 1.05·51-s − 1.29·53-s + 1.76·55-s + 0.362·57-s + 1.08·59-s + 0.0419·61-s + 1.00·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 128 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 160 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 318 T + p^{3} T^{2} \) |
| 41 | \( 1 + 192 T + p^{3} T^{2} \) |
| 43 | \( 1 + 220 T + p^{3} T^{2} \) |
| 47 | \( 1 + 184 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 - 492 T + p^{3} T^{2} \) |
| 61 | \( 1 - 20 T + p^{3} T^{2} \) |
| 67 | \( 1 + 380 T + p^{3} T^{2} \) |
| 71 | \( 1 - 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 560 T + p^{3} T^{2} \) |
| 79 | \( 1 + 104 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1508 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1144 T + p^{3} T^{2} \) |
| 97 | \( 1 + 904 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.633058017059901325551340016680, −7.05995275141344276869021957194, −6.63511351123313250552297716938, −6.02809531780538998209702693248, −5.18709122600757677439807836168, −4.25989048362230228070095627631, −3.42783497724062672754947067290, −1.92363590706641682063118201128, −1.45392321036695402975288099648, 0,
1.45392321036695402975288099648, 1.92363590706641682063118201128, 3.42783497724062672754947067290, 4.25989048362230228070095627631, 5.18709122600757677439807836168, 6.02809531780538998209702693248, 6.63511351123313250552297716938, 7.05995275141344276869021957194, 8.633058017059901325551340016680
