L(s) = 1 | + 2-s − 3-s − 4-s − 2·5-s − 6-s − 2·7-s − 3·8-s − 2·9-s − 2·10-s + 12-s − 13-s − 2·14-s + 2·15-s − 16-s + 7·17-s − 2·18-s − 6·19-s + 2·20-s + 2·21-s − 23-s + 3·24-s − 25-s − 26-s + 5·27-s + 2·28-s − 3·29-s + 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 1.06·8-s − 2/3·9-s − 0.632·10-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s + 1.69·17-s − 0.471·18-s − 1.37·19-s + 0.447·20-s + 0.436·21-s − 0.208·23-s + 0.612·24-s − 1/5·25-s − 0.196·26-s + 0.962·27-s + 0.377·28-s − 0.557·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7090239839\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7090239839\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−9.589501251134218389458509947391, −8.405331371677761313068805876845, −8.023015761347280894276575665357, −6.71450193300509424752038725498, −6.02391114509917305396013448039, −5.30117793036741025472408939150, −4.37356024554280267880826147469, −3.59611860981657522372560116564, −2.77645141796837974948334993714, −0.52542659854360093931758822227,
0.52542659854360093931758822227, 2.77645141796837974948334993714, 3.59611860981657522372560116564, 4.37356024554280267880826147469, 5.30117793036741025472408939150, 6.02391114509917305396013448039, 6.71450193300509424752038725498, 8.023015761347280894276575665357, 8.405331371677761313068805876845, 9.589501251134218389458509947391