| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 13-s − 15-s − 17-s + 8·23-s + 25-s + 27-s − 6·29-s + 4·33-s − 2·37-s + 39-s + 6·41-s + 8·43-s − 45-s − 7·49-s − 51-s − 6·53-s − 4·55-s + 4·59-s − 10·61-s − 65-s − 4·67-s + 8·69-s − 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.696·33-s − 0.328·37-s + 0.160·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s − 49-s − 0.140·51-s − 0.824·53-s − 0.539·55-s + 0.520·59-s − 1.28·61-s − 0.124·65-s − 0.488·67-s + 0.963·69-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.082268568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.082268568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.16745721197070, −14.72835245045164, −14.37227446702401, −13.75347597762516, −13.06010143115307, −12.78571427778046, −12.04317750097838, −11.52347881758163, −10.95367880015867, −10.58026121751187, −9.567222847025757, −9.202383182637825, −8.856922771766001, −8.161166174153456, −7.401226539188099, −7.158415185299624, −6.322241989114367, −5.852071591348858, −4.782809271932475, −4.454239935269341, −3.518719514564674, −3.298190345342090, −2.285373584579012, −1.491134071769676, −0.7030929905248829,
0.7030929905248829, 1.491134071769676, 2.285373584579012, 3.298190345342090, 3.518719514564674, 4.454239935269341, 4.782809271932475, 5.852071591348858, 6.322241989114367, 7.158415185299624, 7.401226539188099, 8.161166174153456, 8.856922771766001, 9.202383182637825, 9.567222847025757, 10.58026121751187, 10.95367880015867, 11.52347881758163, 12.04317750097838, 12.78571427778046, 13.06010143115307, 13.75347597762516, 14.37227446702401, 14.72835245045164, 15.16745721197070
