L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 4·7-s − 8-s + 9-s + 10-s − 12-s + 13-s − 4·14-s + 15-s + 16-s + 2·17-s − 18-s + 4·19-s − 20-s − 4·21-s − 8·23-s + 24-s + 25-s − 26-s − 27-s + 4·28-s − 2·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.872·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.37288692412069, −12.03944405471239, −11.68870715786876, −11.46844126054484, −10.94344960670736, −10.43204963559041, −10.14035627509354, −9.582293458303636, −9.107030592189677, −8.377894745199658, −8.158294227673713, −7.834491268383545, −7.317214078070795, −6.845642202594589, −6.267213179922149, −5.723065387628442, −5.266126829603496, −4.818006855110177, −4.339899242166839, −3.570498573165947, −3.321948340018979, −2.344262563471771, −1.695553631791918, −1.492132307304846, −0.7036077555589020, 0,
0.7036077555589020, 1.492132307304846, 1.695553631791918, 2.344262563471771, 3.321948340018979, 3.570498573165947, 4.339899242166839, 4.818006855110177, 5.266126829603496, 5.723065387628442, 6.267213179922149, 6.845642202594589, 7.317214078070795, 7.834491268383545, 8.158294227673713, 8.377894745199658, 9.107030592189677, 9.582293458303636, 10.14035627509354, 10.43204963559041, 10.94344960670736, 11.46844126054484, 11.68870715786876, 12.03944405471239, 12.37288692412069