| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3.37·7-s + 8-s + 9-s + 10-s + 0.627·11-s + 12-s − 2·13-s + 3.37·14-s + 15-s + 16-s − 4.74·17-s + 18-s − 0.627·19-s + 20-s + 3.37·21-s + 0.627·22-s + 3.37·23-s + 24-s + 25-s − 2·26-s + 27-s + 3.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.27·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.189·11-s + 0.288·12-s − 0.554·13-s + 0.901·14-s + 0.258·15-s + 0.250·16-s − 1.15·17-s + 0.235·18-s − 0.144·19-s + 0.223·20-s + 0.735·21-s + 0.133·22-s + 0.703·23-s + 0.204·24-s + 0.200·25-s − 0.392·26-s + 0.192·27-s + 0.637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.480187251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.480187251\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 + 0.627T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 37 | \( 1 + 0.744T + 37T^{2} \) |
| 41 | \( 1 - 0.744T + 41T^{2} \) |
| 43 | \( 1 - 0.627T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 3.37T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.10122555632368442110937507504, −9.151686987727392263319328945180, −8.383538037439911042697584044446, −7.45183342421001718272130718469, −6.70709419306710233180753915970, −5.48190214137245705985418789142, −4.76158486467898142834850524931, −3.85389128770315371773155464894, −2.50064394170691424181515377433, −1.68007154596548643645809445793,
1.68007154596548643645809445793, 2.50064394170691424181515377433, 3.85389128770315371773155464894, 4.76158486467898142834850524931, 5.48190214137245705985418789142, 6.70709419306710233180753915970, 7.45183342421001718272130718469, 8.383538037439911042697584044446, 9.151686987727392263319328945180, 10.10122555632368442110937507504
