| L(s) = 1 | + 2-s − 3.45·3-s + 4-s − 5-s − 3.45·6-s − 3.54·7-s + 8-s + 8.96·9-s − 10-s + 11-s − 3.45·12-s + 5.54·13-s − 3.54·14-s + 3.45·15-s + 16-s − 6.38·17-s + 8.96·18-s + 7.62·19-s − 20-s + 12.2·21-s + 22-s − 4.31·23-s − 3.45·24-s + 25-s + 5.54·26-s − 20.6·27-s − 3.54·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.99·3-s + 0.5·4-s − 0.447·5-s − 1.41·6-s − 1.33·7-s + 0.353·8-s + 2.98·9-s − 0.316·10-s + 0.301·11-s − 0.998·12-s + 1.53·13-s − 0.947·14-s + 0.893·15-s + 0.250·16-s − 1.54·17-s + 2.11·18-s + 1.74·19-s − 0.223·20-s + 2.67·21-s + 0.213·22-s − 0.899·23-s − 0.706·24-s + 0.200·25-s + 1.08·26-s − 3.96·27-s − 0.669·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013402037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013402037\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 3.45T + 3T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 13 | \( 1 - 5.54T + 13T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 23 | \( 1 + 4.31T + 23T^{2} \) |
| 29 | \( 1 - 2.64T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 2.96T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 0.437T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.037855262721241007713536885556, −7.03223558164473834958196006026, −6.55643771395487621907754261387, −6.18969826771701615039899396710, −5.45840277092474693422281317593, −4.68218760073984666669489083123, −3.89744275825743708982498058913, −3.32906951214174325225585066429, −1.66232235589244256589799960761, −0.56726266658005982415677412700,
0.56726266658005982415677412700, 1.66232235589244256589799960761, 3.32906951214174325225585066429, 3.89744275825743708982498058913, 4.68218760073984666669489083123, 5.45840277092474693422281317593, 6.18969826771701615039899396710, 6.55643771395487621907754261387, 7.03223558164473834958196006026, 8.037855262721241007713536885556
