| L(s) = 1 | + 2-s + 4-s + 2·5-s + 3·7-s + 8-s + 2·10-s + 11-s − 13-s + 3·14-s + 16-s + 8·17-s − 2·19-s + 2·20-s + 22-s + 8·23-s − 25-s − 26-s + 3·28-s + 29-s + 5·31-s + 32-s + 8·34-s + 6·35-s + 6·37-s − 2·38-s + 2·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.13·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s − 0.277·13-s + 0.801·14-s + 1/4·16-s + 1.94·17-s − 0.458·19-s + 0.447·20-s + 0.213·22-s + 1.66·23-s − 1/5·25-s − 0.196·26-s + 0.566·28-s + 0.185·29-s + 0.898·31-s + 0.176·32-s + 1.37·34-s + 1.01·35-s + 0.986·37-s − 0.324·38-s + 0.316·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 223938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.054281479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.054281479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.09886291109265, −12.35405189693104, −12.14849966366842, −11.68244421286932, −11.06921575940217, −10.76740824583262, −10.18853670402453, −9.735016490660686, −9.325247732845931, −8.693726843517155, −8.065767026220511, −7.785319076647674, −7.228927378598866, −6.588282847012334, −6.194287041303849, −5.548155996383632, −5.240334374785196, −4.808724893880282, −4.231212949274065, −3.646324094393014, −2.800221935443838, −2.692620628356238, −1.662552393122022, −1.420588770577979, −0.7555179864567152,
0.7555179864567152, 1.420588770577979, 1.662552393122022, 2.692620628356238, 2.800221935443838, 3.646324094393014, 4.231212949274065, 4.808724893880282, 5.240334374785196, 5.548155996383632, 6.194287041303849, 6.588282847012334, 7.228927378598866, 7.785319076647674, 8.065767026220511, 8.693726843517155, 9.325247732845931, 9.735016490660686, 10.18853670402453, 10.76740824583262, 11.06921575940217, 11.68244421286932, 12.14849966366842, 12.35405189693104, 13.09886291109265
