L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 6·11-s + 5·13-s − 14-s + 16-s − 3·17-s + 19-s + 6·22-s − 3·23-s − 5·25-s + 5·26-s − 28-s − 9·29-s − 4·31-s + 32-s − 3·34-s + 2·37-s + 38-s + 8·43-s + 6·44-s − 3·46-s − 6·49-s − 5·50-s + 5·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.80·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.229·19-s + 1.27·22-s − 0.625·23-s − 25-s + 0.980·26-s − 0.188·28-s − 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.328·37-s + 0.162·38-s + 1.21·43-s + 0.904·44-s − 0.442·46-s − 6/7·49-s − 0.707·50-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.083012379\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.083012379\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−11.50773178855083955803608159036, −10.98048210666359838390472921844, −9.572003756910085560364635840628, −8.872794307339397941412432360955, −7.53492665725128185091116505938, −6.40690082764165790518888770884, −5.85342474444132611353121181432, −4.18290710455057630899406360415, −3.56295849457415925614690839234, −1.70600515778630825145695828253,
1.70600515778630825145695828253, 3.56295849457415925614690839234, 4.18290710455057630899406360415, 5.85342474444132611353121181432, 6.40690082764165790518888770884, 7.53492665725128185091116505938, 8.872794307339397941412432360955, 9.572003756910085560364635840628, 10.98048210666359838390472921844, 11.50773178855083955803608159036