L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 7-s − 8-s + 9-s + 2·11-s + 2·12-s + 13-s − 14-s + 16-s − 2·17-s − 18-s + 2·21-s − 2·22-s + 8·23-s − 2·24-s − 26-s − 4·27-s + 28-s − 2·29-s + 4·31-s − 32-s + 4·33-s + 2·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.577·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.436·21-s − 0.426·22-s + 1.66·23-s − 0.408·24-s − 0.196·26-s − 0.769·27-s + 0.188·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.696·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.361958250\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.361958250\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.512821880469239581420977965099, −7.75304572512655024843087455349, −7.14052770491266167489281588969, −6.39487006354651150868493053523, −5.44671806896628178569427820278, −4.42718675935708633083960418135, −3.54968722621236347894692432259, −2.76635435551339469009684089572, −1.96791033975599744715539526507, −0.931042262026461133004011736748,
0.931042262026461133004011736748, 1.96791033975599744715539526507, 2.76635435551339469009684089572, 3.54968722621236347894692432259, 4.42718675935708633083960418135, 5.44671806896628178569427820278, 6.39487006354651150868493053523, 7.14052770491266167489281588969, 7.75304572512655024843087455349, 8.512821880469239581420977965099