| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s − 3·9-s + 10-s − 4·11-s − 6·13-s + 14-s + 16-s − 2·17-s − 3·18-s + 20-s − 4·22-s + 25-s − 6·26-s + 28-s + 6·29-s + 8·31-s + 32-s − 2·34-s + 35-s − 3·36-s + 10·37-s + 40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.223·20-s − 0.852·22-s + 1/5·25-s − 1.17·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.169·35-s − 1/2·36-s + 1.64·37-s + 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−14.93247724291650, −14.61053102118222, −14.12952821570513, −13.50403181345370, −13.26720233485742, −12.38381889400830, −12.19915084996317, −11.50336128869238, −11.03189411992747, −10.41293563417345, −9.943497161467979, −9.420869790172417, −8.558865567335879, −8.086555385645039, −7.626771856001726, −6.865937886645252, −6.367934368950244, −5.569184832133281, −5.272981212714408, −4.649578310129229, −4.176854322907133, −2.959303010206992, −2.555249746420485, −2.331241819698295, −1.037069227231440, 0,
1.037069227231440, 2.331241819698295, 2.555249746420485, 2.959303010206992, 4.176854322907133, 4.649578310129229, 5.272981212714408, 5.569184832133281, 6.367934368950244, 6.865937886645252, 7.626771856001726, 8.086555385645039, 8.558865567335879, 9.420869790172417, 9.943497161467979, 10.41293563417345, 11.03189411992747, 11.50336128869238, 12.19915084996317, 12.38381889400830, 13.26720233485742, 13.50403181345370, 14.12952821570513, 14.61053102118222, 14.93247724291650
