L(s) = 1 | + 7·3-s − 5·5-s + 22·9-s + 58·11-s + 82·13-s − 35·15-s + 50·17-s + 64·19-s − 111·23-s + 25·25-s − 35·27-s + 103·29-s − 130·31-s + 406·33-s + 376·37-s + 574·39-s − 307·41-s − 197·43-s − 110·45-s + 120·47-s + 350·51-s − 508·53-s − 290·55-s + 448·57-s + 600·59-s − 165·61-s − 410·65-s + ⋯ |
L(s) = 1 | + 1.34·3-s − 0.447·5-s + 0.814·9-s + 1.58·11-s + 1.74·13-s − 0.602·15-s + 0.713·17-s + 0.772·19-s − 1.00·23-s + 1/5·25-s − 0.249·27-s + 0.659·29-s − 0.753·31-s + 2.14·33-s + 1.67·37-s + 2.35·39-s − 1.16·41-s − 0.698·43-s − 0.364·45-s + 0.372·47-s + 0.960·51-s − 1.31·53-s − 0.710·55-s + 1.04·57-s + 1.32·59-s − 0.346·61-s − 0.782·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.469284624\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.469284624\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 58 T + p^{3} T^{2} \) |
| 13 | \( 1 - 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 64 T + p^{3} T^{2} \) |
| 23 | \( 1 + 111 T + p^{3} T^{2} \) |
| 29 | \( 1 - 103 T + p^{3} T^{2} \) |
| 31 | \( 1 + 130 T + p^{3} T^{2} \) |
| 37 | \( 1 - 376 T + p^{3} T^{2} \) |
| 41 | \( 1 + 307 T + p^{3} T^{2} \) |
| 43 | \( 1 + 197 T + p^{3} T^{2} \) |
| 47 | \( 1 - 120 T + p^{3} T^{2} \) |
| 53 | \( 1 + 508 T + p^{3} T^{2} \) |
| 59 | \( 1 - 600 T + p^{3} T^{2} \) |
| 61 | \( 1 + 165 T + p^{3} T^{2} \) |
| 67 | \( 1 + 633 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 606 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1316 T + p^{3} T^{2} \) |
| 83 | \( 1 - 61 T + p^{3} T^{2} \) |
| 89 | \( 1 + 187 T + p^{3} T^{2} \) |
| 97 | \( 1 - 406 T + p^{3} T^{2} \) |
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
−8.676553474377649942673471848582, −8.247943147732702306971645742872, −7.47996183666778713478632475083, −6.52630658385393580761970481981, −5.79118231618208177825124364375, −4.35695642566528996713525636995, −3.61951646702961242253018422200, −3.22140876217819600507664522674, −1.81462880685168418855658246718, −0.998052417694111841043203596599,
0.998052417694111841043203596599, 1.81462880685168418855658246718, 3.22140876217819600507664522674, 3.61951646702961242253018422200, 4.35695642566528996713525636995, 5.79118231618208177825124364375, 6.52630658385393580761970481981, 7.47996183666778713478632475083, 8.247943147732702306971645742872, 8.676553474377649942673471848582