L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 7-s − 3·8-s + 9-s + 10-s − 12-s − 14-s + 15-s − 16-s − 17-s + 18-s − 4·19-s − 20-s − 21-s − 3·24-s + 25-s + 27-s + 28-s − 2·29-s + 30-s + 5·32-s − 34-s − 35-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.182·30-s + 0.883·32-s − 0.171·34-s − 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.04831538694839, −12.58964276330028, −12.28518236744292, −11.64385715360382, −11.05885514794304, −10.63760549181492, −10.08717221538863, −9.566383442231204, −9.176714796047980, −8.986365536577751, −8.240279045854593, −7.953822115629301, −7.299020735706699, −6.643094898886474, −6.336471368691943, −5.657644896408693, −5.490892914989544, −4.618792282090078, −4.187906772848646, −4.026899689318227, −3.110244345871055, −2.827385201178811, −2.273518298844255, −1.562292962542364, −0.7733069414217003, 0,
0.7733069414217003, 1.562292962542364, 2.273518298844255, 2.827385201178811, 3.110244345871055, 4.026899689318227, 4.187906772848646, 4.618792282090078, 5.490892914989544, 5.657644896408693, 6.336471368691943, 6.643094898886474, 7.299020735706699, 7.953822115629301, 8.240279045854593, 8.986365536577751, 9.176714796047980, 9.566383442231204, 10.08717221538863, 10.63760549181492, 11.05885514794304, 11.64385715360382, 12.28518236744292, 12.58964276330028, 13.04831538694839