L(s) = 1 | − 2-s + 1.50·3-s + 4-s − 2.98·5-s − 1.50·6-s − 3.73·7-s − 8-s − 0.745·9-s + 2.98·10-s − 3.81·11-s + 1.50·12-s + 3.73·14-s − 4.48·15-s + 16-s − 2.89·17-s + 0.745·18-s − 19-s − 2.98·20-s − 5.60·21-s + 3.81·22-s − 1.66·23-s − 1.50·24-s + 3.93·25-s − 5.62·27-s − 3.73·28-s − 8.51·29-s + 4.48·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.866·3-s + 0.5·4-s − 1.33·5-s − 0.613·6-s − 1.41·7-s − 0.353·8-s − 0.248·9-s + 0.945·10-s − 1.15·11-s + 0.433·12-s + 0.997·14-s − 1.15·15-s + 0.250·16-s − 0.701·17-s + 0.175·18-s − 0.229·19-s − 0.668·20-s − 1.22·21-s + 0.813·22-s − 0.348·23-s − 0.306·24-s + 0.786·25-s − 1.08·27-s − 0.705·28-s − 1.58·29-s + 0.819·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01069160939\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01069160939\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 + 8.51T + 29T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 5.42T + 43T^{2} \) |
| 47 | \( 1 - 6.13T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 + 5.99T + 61T^{2} \) |
| 67 | \( 1 - 7.35T + 67T^{2} \) |
| 71 | \( 1 - 9.75T + 71T^{2} \) |
| 73 | \( 1 - 0.118T + 73T^{2} \) |
| 79 | \( 1 - 3.04T + 79T^{2} \) |
| 83 | \( 1 - 1.22T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.075797486107401943614378342817, −7.52354875127224734577016912891, −6.93046259707544768132393665746, −6.10069891722196328214975027000, −5.21881073696739736513141367560, −4.01281019582678976678257482697, −3.43418358730874354843207447436, −2.85233043911770817073957635857, −1.99181322904599126754318785834, −0.05198208272884298449074385928,
0.05198208272884298449074385928, 1.99181322904599126754318785834, 2.85233043911770817073957635857, 3.43418358730874354843207447436, 4.01281019582678976678257482697, 5.21881073696739736513141367560, 6.10069891722196328214975027000, 6.93046259707544768132393665746, 7.52354875127224734577016912891, 8.075797486107401943614378342817