| L(s) = 1 | + 3-s − 2·4-s + 3·5-s + 4·7-s + 9-s − 2·12-s + 13-s + 3·15-s + 4·16-s + 17-s + 19-s − 6·20-s + 4·21-s + 9·23-s + 4·25-s + 27-s − 8·28-s − 6·29-s + 2·31-s + 12·35-s − 2·36-s − 4·37-s + 39-s + 3·41-s + 7·43-s + 3·45-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s + 1.51·7-s + 1/3·9-s − 0.577·12-s + 0.277·13-s + 0.774·15-s + 16-s + 0.242·17-s + 0.229·19-s − 1.34·20-s + 0.872·21-s + 1.87·23-s + 4/5·25-s + 0.192·27-s − 1.51·28-s − 1.11·29-s + 0.359·31-s + 2.02·35-s − 1/3·36-s − 0.657·37-s + 0.160·39-s + 0.468·41-s + 1.06·43-s + 0.447·45-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.532526180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.532526180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.123063877468170217097057069948, −7.60195787768679259689608146982, −6.65141393178900011874126957740, −5.62535288746366565403300368914, −5.18754379686639968375671320234, −4.59411129964790589621534187851, −3.66589986463958216194059753670, −2.69595897502735310420716823107, −1.70457407581975008871008195419, −1.09494185617481664600047301525,
1.09494185617481664600047301525, 1.70457407581975008871008195419, 2.69595897502735310420716823107, 3.66589986463958216194059753670, 4.59411129964790589621534187851, 5.18754379686639968375671320234, 5.62535288746366565403300368914, 6.65141393178900011874126957740, 7.60195787768679259689608146982, 8.123063877468170217097057069948
