L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 6·11-s − 4·13-s − 14-s + 16-s − 2·17-s + 20-s − 6·22-s − 23-s + 25-s + 4·26-s + 28-s + 8·29-s + 4·31-s − 32-s + 2·34-s + 35-s − 6·37-s − 40-s + 10·41-s + 6·43-s + 6·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s − 1.27·22-s − 0.208·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.48·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.986·37-s − 0.158·40-s + 1.56·41-s + 0.914·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.014692364\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.014692364\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.16402447626489, −15.65563120123140, −14.95094103840675, −14.35226125892822, −14.13577446614954, −13.40737788730392, −12.46675286302481, −12.00800115345956, −11.74128667893912, −10.82654107057093, −10.40133215063148, −9.718897731588556, −9.128179898678271, −8.846083874796828, −8.046510851316325, −7.305813939504716, −6.811833261442633, −6.206856812382179, −5.573473453372467, −4.527380686740836, −4.192764027244083, −3.013465283944821, −2.330032244730638, −1.508289974869421, −0.7381052180322679,
0.7381052180322679, 1.508289974869421, 2.330032244730638, 3.013465283944821, 4.192764027244083, 4.527380686740836, 5.573473453372467, 6.206856812382179, 6.811833261442633, 7.305813939504716, 8.046510851316325, 8.846083874796828, 9.128179898678271, 9.718897731588556, 10.40133215063148, 10.82654107057093, 11.74128667893912, 12.00800115345956, 12.46675286302481, 13.40737788730392, 14.13577446614954, 14.35226125892822, 14.95094103840675, 15.65563120123140, 16.16402447626489