| L(s) = 1 | + 2-s + 4-s + 5·7-s + 8-s + 4·11-s + 13-s + 5·14-s + 16-s − 3·17-s + 19-s + 4·22-s + 7·23-s + 26-s + 5·28-s + 3·29-s − 2·31-s + 32-s − 3·34-s + 2·37-s + 38-s + 6·41-s − 6·43-s + 4·44-s + 7·46-s + 18·49-s + 52-s − 13·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.88·7-s + 0.353·8-s + 1.20·11-s + 0.277·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s + 0.229·19-s + 0.852·22-s + 1.45·23-s + 0.196·26-s + 0.944·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s − 0.514·34-s + 0.328·37-s + 0.162·38-s + 0.937·41-s − 0.914·43-s + 0.603·44-s + 1.03·46-s + 18/7·49-s + 0.138·52-s − 1.78·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.136091102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.136091102\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.64412675368342072100402643347, −7.07704541117290869170938897697, −6.35430569173332358701577738897, −5.60181065616457415736945006525, −4.73764288830413259853760333013, −4.52726401175650390589229310145, −3.63706267746930921054569078234, −2.68059787783471348034578632155, −1.68490358819888880085777025199, −1.14285782158350026604192915140,
1.14285782158350026604192915140, 1.68490358819888880085777025199, 2.68059787783471348034578632155, 3.63706267746930921054569078234, 4.52726401175650390589229310145, 4.73764288830413259853760333013, 5.60181065616457415736945006525, 6.35430569173332358701577738897, 7.07704541117290869170938897697, 7.64412675368342072100402643347
