| L(s) = 1 | + 5·5-s + 11.8·7-s + 56.2·11-s + 34.5·13-s + 39.2·17-s + 146.·19-s + 23.5·23-s + 25·25-s + 161.·29-s + 29.5·31-s + 59.0·35-s − 217.·37-s + 142.·41-s + 468.·43-s − 394.·47-s − 203.·49-s − 134.·53-s + 281.·55-s + 131.·59-s + 259.·61-s + 172.·65-s − 445.·67-s − 560.·71-s − 88.6·73-s + 663.·77-s − 450.·79-s − 284.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.637·7-s + 1.54·11-s + 0.738·13-s + 0.560·17-s + 1.76·19-s + 0.213·23-s + 0.200·25-s + 1.03·29-s + 0.171·31-s + 0.285·35-s − 0.967·37-s + 0.541·41-s + 1.65·43-s − 1.22·47-s − 0.593·49-s − 0.349·53-s + 0.689·55-s + 0.289·59-s + 0.545·61-s + 0.330·65-s − 0.811·67-s − 0.937·71-s − 0.142·73-s + 0.982·77-s − 0.641·79-s − 0.375·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.789223598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.789223598\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 11.8T + 343T^{2} \) |
| 11 | \( 1 - 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 23.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 161.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 29.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 468.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 131.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 259.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 445.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 560.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 450.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 284.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 625.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 193.T + 9.12e5T^{2} \) |
| 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
−8.823933466469988643353232401355, −7.976274005697138749392763085576, −7.14102993843558305942953036009, −6.34320096136471952322337072524, −5.59576874861547462220902075233, −4.73877211429968613434398865026, −3.78392473397953124102140925707, −2.92967343352485972025345683551, −1.50661475474859047616749923635, −1.04091342433469792579591128801,
1.04091342433469792579591128801, 1.50661475474859047616749923635, 2.92967343352485972025345683551, 3.78392473397953124102140925707, 4.73877211429968613434398865026, 5.59576874861547462220902075233, 6.34320096136471952322337072524, 7.14102993843558305942953036009, 7.976274005697138749392763085576, 8.823933466469988643353232401355
