| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 3·11-s + 13-s + 16-s + 8·17-s + 20-s − 3·22-s − 4·23-s − 4·25-s − 26-s + 29-s + 3·31-s − 32-s − 8·34-s + 8·37-s − 40-s + 2·41-s − 11·43-s + 3·44-s + 4·46-s + 13·47-s + 4·50-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.904·11-s + 0.277·13-s + 1/4·16-s + 1.94·17-s + 0.223·20-s − 0.639·22-s − 0.834·23-s − 4/5·25-s − 0.196·26-s + 0.185·29-s + 0.538·31-s − 0.176·32-s − 1.37·34-s + 1.31·37-s − 0.158·40-s + 0.312·41-s − 1.67·43-s + 0.452·44-s + 0.589·46-s + 1.89·47-s + 0.565·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.254643628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.254643628\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.27132797522191, −14.85171635559052, −14.35241889137183, −13.68290146185541, −13.40461613714866, −12.35276316386749, −12.05477066660577, −11.65853681907519, −10.90833948439518, −10.22451044388613, −9.853077890086352, −9.468253578569568, −8.717942719506083, −8.194869595427934, −7.634366313706251, −7.066481343792587, −6.276835699721522, −5.862302239689648, −5.343571884950017, −4.263931782104213, −3.724575082656816, −2.948583605041258, −2.139638118179713, −1.354266725180875, −0.7304292330032570,
0.7304292330032570, 1.354266725180875, 2.139638118179713, 2.948583605041258, 3.724575082656816, 4.263931782104213, 5.343571884950017, 5.862302239689648, 6.276835699721522, 7.066481343792587, 7.634366313706251, 8.194869595427934, 8.717942719506083, 9.468253578569568, 9.853077890086352, 10.22451044388613, 10.90833948439518, 11.65853681907519, 12.05477066660577, 12.35276316386749, 13.40461613714866, 13.68290146185541, 14.35241889137183, 14.85171635559052, 15.27132797522191
