| L(s) = 1 | − 5·5-s − 18.4·7-s + 55.3·11-s + 52·13-s − 66·17-s + 36.8·23-s + 25·25-s + 46·29-s − 258.·31-s + 92.1·35-s − 24·37-s − 72·41-s − 331.·47-s − 3·49-s + 62·53-s − 276.·55-s − 239.·59-s − 190·61-s − 260·65-s + 774.·67-s − 626.·71-s + 1.07e3·73-s − 1.02e3·77-s + 848.·79-s + 774.·83-s + 330·85-s + 1.11e3·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.995·7-s + 1.51·11-s + 1.10·13-s − 0.941·17-s + 0.334·23-s + 0.200·25-s + 0.294·29-s − 1.49·31-s + 0.445·35-s − 0.106·37-s − 0.274·41-s − 1.03·47-s − 0.00874·49-s + 0.160·53-s − 0.678·55-s − 0.528·59-s − 0.398·61-s − 0.496·65-s + 1.41·67-s − 1.04·71-s + 1.72·73-s − 1.50·77-s + 1.20·79-s + 1.02·83-s + 0.421·85-s + 1.32·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.692155108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.692155108\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 18.4T + 343T^{2} \) |
| 11 | \( 1 - 55.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52T + 2.19e3T^{2} \) |
| 17 | \( 1 + 66T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 - 36.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 46T + 2.43e4T^{2} \) |
| 31 | \( 1 + 258.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 24T + 5.06e4T^{2} \) |
| 41 | \( 1 + 72T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 331.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 62T + 1.48e5T^{2} \) |
| 59 | \( 1 + 239.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 190T + 2.26e5T^{2} \) |
| 67 | \( 1 - 774.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 626.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 848.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 774.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 834T + 9.12e5T^{2} \) |
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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
−9.068698514318895835479275763756, −8.607948125563137605126435209120, −7.44038854039354721481316222609, −6.51978728412169181921754885038, −6.24395935838275332120558185399, −4.87941460860084438545567647678, −3.78497038272051989190693286804, −3.38871306543739995311897237341, −1.86521474621137072072686589574, −0.64655032527873099175723064487,
0.64655032527873099175723064487, 1.86521474621137072072686589574, 3.38871306543739995311897237341, 3.78497038272051989190693286804, 4.87941460860084438545567647678, 6.24395935838275332120558185399, 6.51978728412169181921754885038, 7.44038854039354721481316222609, 8.607948125563137605126435209120, 9.068698514318895835479275763756
