| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s + 13-s − 4·14-s + 16-s − 4·19-s − 20-s + 6·23-s + 25-s − 26-s + 4·28-s − 6·29-s + 10·31-s − 32-s − 4·35-s + 10·37-s + 4·38-s + 40-s − 6·41-s − 4·43-s − 6·46-s − 12·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s − 1.11·29-s + 1.79·31-s − 0.176·32-s − 0.676·35-s + 1.64·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.884·46-s − 1.75·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.090110580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.090110580\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.71249617903290, −11.78143113732152, −11.52876138828428, −11.16702599253505, −11.01552029812973, −10.30410579141158, −9.759722768262504, −9.510828826600156, −8.629218193062347, −8.377756538908056, −8.193844272595992, −7.669308946093110, −7.112213465812387, −6.550051518110642, −6.315482547286066, −5.346463077853024, −5.109883996784280, −4.547860045157504, −4.041344016218470, −3.441735385892960, −2.742625322776204, −2.238758590364277, −1.585029083487440, −1.125794367134388, −0.4610168907306440,
0.4610168907306440, 1.125794367134388, 1.585029083487440, 2.238758590364277, 2.742625322776204, 3.441735385892960, 4.041344016218470, 4.547860045157504, 5.109883996784280, 5.346463077853024, 6.315482547286066, 6.550051518110642, 7.112213465812387, 7.669308946093110, 8.193844272595992, 8.377756538908056, 8.629218193062347, 9.510828826600156, 9.759722768262504, 10.30410579141158, 11.01552029812973, 11.16702599253505, 11.52876138828428, 11.78143113732152, 12.71249617903290
