| L(s) = 1 | + 3·3-s − 19·7-s + 9·9-s + 22·11-s + 13-s − 58·17-s − 53·19-s − 57·21-s + 58·23-s + 27·27-s + 22·29-s − 35·31-s + 66·33-s − 270·37-s + 3·39-s − 468·41-s − 431·43-s − 230·47-s + 18·49-s − 174·51-s − 159·57-s + 446·59-s + 127·61-s − 171·63-s − 811·67-s + 174·69-s + 36·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.02·7-s + 1/3·9-s + 0.603·11-s + 0.0213·13-s − 0.827·17-s − 0.639·19-s − 0.592·21-s + 0.525·23-s + 0.192·27-s + 0.140·29-s − 0.202·31-s + 0.348·33-s − 1.19·37-s + 0.0123·39-s − 1.78·41-s − 1.52·43-s − 0.713·47-s + 0.0524·49-s − 0.477·51-s − 0.369·57-s + 0.984·59-s + 0.266·61-s − 0.341·63-s − 1.47·67-s + 0.303·69-s + 0.0601·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 - T + p^{3} T^{2} \) |
| 17 | \( 1 + 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 53 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 22 T + p^{3} T^{2} \) |
| 31 | \( 1 + 35 T + p^{3} T^{2} \) |
| 37 | \( 1 + 270 T + p^{3} T^{2} \) |
| 41 | \( 1 + 468 T + p^{3} T^{2} \) |
| 43 | \( 1 + 431 T + p^{3} T^{2} \) |
| 47 | \( 1 + 230 T + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 - 446 T + p^{3} T^{2} \) |
| 61 | \( 1 - 127 T + p^{3} T^{2} \) |
| 67 | \( 1 + 811 T + p^{3} T^{2} \) |
| 71 | \( 1 - 36 T + p^{3} T^{2} \) |
| 73 | \( 1 - 522 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1368 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1138 T + p^{3} T^{2} \) |
| 89 | \( 1 - 144 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1079 T + p^{3} T^{2} \) |
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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
−9.749464304629879540523072447929, −8.937882788002441368493599278965, −8.275377707739904371508814221724, −6.87096297174694726630498678436, −6.55052805163720812530713994647, −5.10516993583275439241068301798, −3.90193928019172518264515322207, −3.05305308136204721753667325825, −1.74974125783886608431223465159, 0,
1.74974125783886608431223465159, 3.05305308136204721753667325825, 3.90193928019172518264515322207, 5.10516993583275439241068301798, 6.55052805163720812530713994647, 6.87096297174694726630498678436, 8.275377707739904371508814221724, 8.937882788002441368493599278965, 9.749464304629879540523072447929
