| L(s) = 1 | − 2-s + 4-s + 5-s − 3·7-s − 8-s − 10-s + 11-s + 13-s + 3·14-s + 16-s − 17-s + 19-s + 20-s − 22-s − 5·23-s + 25-s − 26-s − 3·28-s + 4·29-s − 4·31-s − 32-s + 34-s − 3·35-s + 37-s − 38-s − 40-s + 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.13·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 0.223·20-s − 0.213·22-s − 1.04·23-s + 1/5·25-s − 0.196·26-s − 0.566·28-s + 0.742·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s − 0.507·35-s + 0.164·37-s − 0.162·38-s − 0.158·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.362249881485377076415919875017, −7.55034026338207995415389319230, −6.68217966129043755752514948264, −6.23113354947560640054436928072, −5.46996779571125576249027450649, −4.24177656555103378057100668816, −3.32545598840517938623926504966, −2.47033568366941202377811817806, −1.36722901411608161833416969920, 0,
1.36722901411608161833416969920, 2.47033568366941202377811817806, 3.32545598840517938623926504966, 4.24177656555103378057100668816, 5.46996779571125576249027450649, 6.23113354947560640054436928072, 6.68217966129043755752514948264, 7.55034026338207995415389319230, 8.362249881485377076415919875017
