| L(s) = 1 | + 4.30·3-s − 11.4·5-s + 16.0·7-s − 8.49·9-s − 3.61·11-s − 13·13-s − 49.4·15-s − 132.·17-s + 33.5·19-s + 69.1·21-s + 99.6·23-s + 7.09·25-s − 152.·27-s + 177.·29-s − 342.·31-s − 15.5·33-s − 184.·35-s − 258.·37-s − 55.9·39-s − 164.·41-s − 424.·43-s + 97.6·45-s + 327.·47-s − 84.4·49-s − 571.·51-s + 575.·53-s + 41.5·55-s + ⋯ |
| L(s) = 1 | + 0.827·3-s − 1.02·5-s + 0.868·7-s − 0.314·9-s − 0.0991·11-s − 0.277·13-s − 0.851·15-s − 1.89·17-s + 0.404·19-s + 0.718·21-s + 0.903·23-s + 0.0567·25-s − 1.08·27-s + 1.13·29-s − 1.98·31-s − 0.0820·33-s − 0.892·35-s − 1.14·37-s − 0.229·39-s − 0.626·41-s − 1.50·43-s + 0.323·45-s + 1.01·47-s − 0.246·49-s − 1.56·51-s + 1.49·53-s + 0.101·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 - 4.30T + 27T^{2} \) |
| 5 | \( 1 + 11.4T + 125T^{2} \) |
| 7 | \( 1 - 16.0T + 343T^{2} \) |
| 11 | \( 1 + 3.61T + 1.33e3T^{2} \) |
| 17 | \( 1 + 132.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 177.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 342.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 164.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 424.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 327.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 575.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 77.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 231.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.16T + 3.00e5T^{2} \) |
| 71 | \( 1 + 596.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 260.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 970.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 278.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 811.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 10.1T + 9.12e5T^{2} \) |
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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
−10.48653300056634810901467735807, −8.973259684387320450364451912628, −8.624248266666682589140678975018, −7.67071207961558935680113560275, −6.90114940467134700708324747503, −5.26827849963050016320447577284, −4.27976415589290371948916369255, −3.19795042609707651852271583994, −1.95509575559079745576220730456, 0,
1.95509575559079745576220730456, 3.19795042609707651852271583994, 4.27976415589290371948916369255, 5.26827849963050016320447577284, 6.90114940467134700708324747503, 7.67071207961558935680113560275, 8.624248266666682589140678975018, 8.973259684387320450364451912628, 10.48653300056634810901467735807
