| L(s) = 1 | + 0.229·2-s − 7.94·4-s − 5·5-s − 3.66·8-s − 1.14·10-s − 51.0·11-s − 46.0·13-s + 62.7·16-s − 72.7·17-s + 123.·19-s + 39.7·20-s − 11.7·22-s + 156.·23-s + 25·25-s − 10.5·26-s + 191.·29-s + 116.·31-s + 43.7·32-s − 16.7·34-s + 83.1·37-s + 28.4·38-s + 18.3·40-s − 466.·41-s + 422.·43-s + 405.·44-s + 35.8·46-s − 268.·47-s + ⋯ |
| L(s) = 1 | + 0.0812·2-s − 0.993·4-s − 0.447·5-s − 0.161·8-s − 0.0363·10-s − 1.39·11-s − 0.983·13-s + 0.980·16-s − 1.03·17-s + 1.49·19-s + 0.444·20-s − 0.113·22-s + 1.41·23-s + 0.200·25-s − 0.0798·26-s + 1.22·29-s + 0.673·31-s + 0.241·32-s − 0.0843·34-s + 0.369·37-s + 0.121·38-s + 0.0724·40-s − 1.77·41-s + 1.49·43-s + 1.38·44-s + 0.114·46-s − 0.833·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.229T + 8T^{2} \) |
| 11 | \( 1 + 51.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 466.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 268.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 709.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 402.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 114.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 214.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 402.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 366.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
| 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
−8.404322031895261982085747467458, −7.56007168230109525199202024175, −6.98034027845875944753028343484, −5.68070432560536202912531800905, −4.90355854984710791236358753958, −4.56164276561303067927886998633, −3.25404190451837747551778945480, −2.60613219378183874690377964090, −0.934201295836527540725512708229, 0,
0.934201295836527540725512708229, 2.60613219378183874690377964090, 3.25404190451837747551778945480, 4.56164276561303067927886998633, 4.90355854984710791236358753958, 5.68070432560536202912531800905, 6.98034027845875944753028343484, 7.56007168230109525199202024175, 8.404322031895261982085747467458
