| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 3·9-s − 10-s + 2·13-s + 4·14-s + 16-s + 6·17-s − 3·18-s + 4·19-s − 20-s − 6·23-s + 25-s + 2·26-s + 4·28-s − 29-s − 4·31-s + 32-s + 6·34-s − 4·35-s − 3·36-s + 8·37-s + 4·38-s − 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 9-s − 0.316·10-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 0.185·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.676·35-s − 1/2·36-s + 1.31·37-s + 0.648·38-s − 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.537849381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.537849381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.67599962251875, −14.31492684085392, −13.94321830843210, −13.29545424864861, −12.70775561161974, −11.90497353276651, −11.63063306950775, −11.47341746516731, −10.77232421391699, −10.16585318808251, −9.561609588022876, −8.720958678484791, −8.116451087225944, −7.938567654565086, −7.378054090998305, −6.545297352891634, −5.826712280975275, −5.378786057085160, −5.012284266932806, −4.180970319747372, −3.608877475224731, −3.089647400027983, −2.197440530188420, −1.524901207363430, −0.7129416753421461,
0.7129416753421461, 1.524901207363430, 2.197440530188420, 3.089647400027983, 3.608877475224731, 4.180970319747372, 5.012284266932806, 5.378786057085160, 5.826712280975275, 6.545297352891634, 7.378054090998305, 7.938567654565086, 8.116451087225944, 8.720958678484791, 9.561609588022876, 10.16585318808251, 10.77232421391699, 11.47341746516731, 11.63063306950775, 11.90497353276651, 12.70775561161974, 13.29545424864861, 13.94321830843210, 14.31492684085392, 14.67599962251875
