| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 11-s + 3·13-s + 14-s + 16-s − 8·17-s − 3·19-s + 20-s + 22-s + 6·23-s + 25-s − 3·26-s − 28-s − 6·29-s − 4·31-s − 32-s + 8·34-s − 35-s + 2·37-s + 3·38-s − 40-s − 11·41-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 − 0.301·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 1.94·17-s − 0.688·19-s + 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s − 0.588·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.37·34-s − 0.169·35-s + 0.328·37-s + 0.486·38-s − 0.158·40-s − 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−8.950147329144740287143351852138, −8.277510386844781328548026211615, −7.15446785582654008177329147897, −6.59995338126477653770083404421, −5.83194924670353997323190002178, −4.79042522916138426389388722017, −3.66952951517574238772185308380, −2.56610622763550007370416436879, −1.61254885304796420568221989379, 0,
1.61254885304796420568221989379, 2.56610622763550007370416436879, 3.66952951517574238772185308380, 4.79042522916138426389388722017, 5.83194924670353997323190002178, 6.59995338126477653770083404421, 7.15446785582654008177329147897, 8.277510386844781328548026211615, 8.950147329144740287143351852138
