L(s) = 1 | − 0.860·2-s − 3-s − 1.25·4-s − 1.02·5-s + 0.860·6-s − 2.10·7-s + 2.80·8-s + 9-s + 0.882·10-s − 11-s + 1.25·12-s + 0.448·13-s + 1.81·14-s + 1.02·15-s + 0.105·16-s + 6.98·17-s − 0.860·18-s + 2.35·19-s + 1.29·20-s + 2.10·21-s + 0.860·22-s − 9.42·23-s − 2.80·24-s − 3.94·25-s − 0.385·26-s − 27-s + 2.64·28-s + ⋯ |
L(s) = 1 | − 0.608·2-s − 0.577·3-s − 0.629·4-s − 0.458·5-s + 0.351·6-s − 0.795·7-s + 0.991·8-s + 0.333·9-s + 0.279·10-s − 0.301·11-s + 0.363·12-s + 0.124·13-s + 0.483·14-s + 0.264·15-s + 0.0263·16-s + 1.69·17-s − 0.202·18-s + 0.540·19-s + 0.288·20-s + 0.459·21-s + 0.183·22-s − 1.96·23-s − 0.572·24-s − 0.789·25-s − 0.0756·26-s − 0.192·27-s + 0.500·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4523246635\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4523246635\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 97 | \( 1 - T \) |
good | 2 | \( 1 + 0.860T + 2T^{2} \) |
| 5 | \( 1 + 1.02T + 5T^{2} \) |
| 7 | \( 1 + 2.10T + 7T^{2} \) |
| 13 | \( 1 - 0.448T + 13T^{2} \) |
| 17 | \( 1 - 6.98T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 + 9.42T + 23T^{2} \) |
| 29 | \( 1 - 1.98T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 + 8.78T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 + 3.78T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 - 6.14T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 1.63T + 83T^{2} \) |
| 89 | \( 1 + 2.24T + 89T^{2} \) |
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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
−8.492369093712464063808885335441, −8.042474380417479516588219196516, −7.36228743183366837178386650519, −6.42520603200996393101316043237, −5.61013781708330819507319893433, −4.91607970476351571069566157822, −3.88008920186257163335068445057, −3.29873487247235224668779989049, −1.67501091904341700344880173333, −0.46756935762397751828706837141,
0.46756935762397751828706837141, 1.67501091904341700344880173333, 3.29873487247235224668779989049, 3.88008920186257163335068445057, 4.91607970476351571069566157822, 5.61013781708330819507319893433, 6.42520603200996393101316043237, 7.36228743183366837178386650519, 8.042474380417479516588219196516, 8.492369093712464063808885335441