| L(s) = 1 | − 2-s + 4-s − 5-s − 5·7-s − 8-s + 10-s + 3·11-s + 5·14-s + 16-s − 17-s − 2·19-s − 20-s − 3·22-s + 9·23-s + 25-s − 5·28-s − 6·29-s − 8·31-s − 32-s + 34-s + 5·35-s − 2·37-s + 2·38-s + 40-s + 12·41-s − 43-s + 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.88·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s + 1.33·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s − 0.223·20-s − 0.639·22-s + 1.87·23-s + 1/5·25-s − 0.944·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.171·34-s + 0.845·35-s − 0.328·37-s + 0.324·38-s + 0.158·40-s + 1.87·41-s − 0.152·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5888844018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5888844018\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−12.83576240185622, −12.40744073715497, −11.90810263709202, −11.31580750763606, −10.76709105838460, −10.74484434244825, −9.811809599148979, −9.512868219626044, −9.138808397048991, −8.838304411511987, −8.310221434836516, −7.354472610688419, −7.265498818024954, −6.789760984568036, −6.363034342997272, −5.739823301236841, −5.403736961563256, −4.442689758762808, −3.842713690080937, −3.613415576408483, −2.858115930549146, −2.556284091715979, −1.637322389155157, −0.9673449344128875, −0.2766016886887912,
0.2766016886887912, 0.9673449344128875, 1.637322389155157, 2.556284091715979, 2.858115930549146, 3.613415576408483, 3.842713690080937, 4.442689758762808, 5.403736961563256, 5.739823301236841, 6.363034342997272, 6.789760984568036, 7.265498818024954, 7.354472610688419, 8.310221434836516, 8.838304411511987, 9.138808397048991, 9.512868219626044, 9.811809599148979, 10.74484434244825, 10.76709105838460, 11.31580750763606, 11.90810263709202, 12.40744073715497, 12.83576240185622
