| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s + 14-s + 16-s + 5·17-s − 20-s − 22-s + 23-s + 25-s − 28-s − 5·29-s − 2·31-s − 32-s − 5·34-s + 35-s + 2·37-s + 40-s + 2·41-s − 9·43-s + 44-s − 46-s + 3·47-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 + 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.223·20-s − 0.213·22-s + 0.208·23-s + 1/5·25-s − 0.188·28-s − 0.928·29-s − 0.359·31-s − 0.176·32-s − 0.857·34-s + 0.169·35-s + 0.328·37-s + 0.158·40-s + 0.312·41-s − 1.37·43-s + 0.150·44-s − 0.147·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.417595490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417595490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.38657248673625, −12.07055672675098, −11.49860719343133, −11.21942797849005, −10.72114524463092, −10.15231285049864, −9.771781731163787, −9.400595952634623, −8.959210202091199, −8.288950044027749, −8.063407755806850, −7.513038963402178, −7.085560827076522, −6.606952029336444, −6.148376761099967, −5.504318930452057, −5.171370084220488, −4.454092784439460, −3.773637940108475, −3.422014244494356, −2.967355902500200, −2.188352574357772, −1.663023109159934, −0.9583495258927134, −0.4102097342419255,
0.4102097342419255, 0.9583495258927134, 1.663023109159934, 2.188352574357772, 2.967355902500200, 3.422014244494356, 3.773637940108475, 4.454092784439460, 5.171370084220488, 5.504318930452057, 6.148376761099967, 6.606952029336444, 7.085560827076522, 7.513038963402178, 8.063407755806850, 8.288950044027749, 8.959210202091199, 9.400595952634623, 9.771781731163787, 10.15231285049864, 10.72114524463092, 11.21942797849005, 11.49860719343133, 12.07055672675098, 12.38657248673625
