| L(s) = 1 | + 5.37·3-s − 11.8·5-s − 26.4·7-s + 1.86·9-s + 49.8·11-s + 49.0·13-s − 63.7·15-s + 17.2·17-s + 19·19-s − 142.·21-s + 166.·23-s + 15.6·25-s − 135.·27-s + 109.·29-s − 273.·31-s + 267.·33-s + 314.·35-s − 167.·37-s + 263.·39-s + 15.1·41-s − 413.·43-s − 22.0·45-s − 161.·47-s + 358.·49-s + 92.5·51-s + 490.·53-s − 591.·55-s + ⋯ |
| L(s) = 1 | + 1.03·3-s − 1.06·5-s − 1.43·7-s + 0.0689·9-s + 1.36·11-s + 1.04·13-s − 1.09·15-s + 0.245·17-s + 0.229·19-s − 1.47·21-s + 1.51·23-s + 0.125·25-s − 0.962·27-s + 0.699·29-s − 1.58·31-s + 1.41·33-s + 1.51·35-s − 0.742·37-s + 1.08·39-s + 0.0577·41-s − 1.46·43-s − 0.0731·45-s − 0.501·47-s + 1.04·49-s + 0.254·51-s + 1.27·53-s − 1.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 5.37T + 27T^{2} \) |
| 5 | \( 1 + 11.8T + 125T^{2} \) |
| 7 | \( 1 + 26.4T + 343T^{2} \) |
| 11 | \( 1 - 49.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 49.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.2T + 4.91e3T^{2} \) |
| 23 | \( 1 - 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 15.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 335.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 497.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 798.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 311.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 665.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 372.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 960.T + 9.12e5T^{2} \) |
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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
−8.946069198867718019916370215626, −8.349946623726534446920066140609, −7.25646776305200764167941850661, −6.70631343420272650918033110902, −5.69028292856084874870147978156, −4.15944903477114194023122068929, −3.44520716762330185605204366758, −3.08138258551269292699289428468, −1.39475089522996620097633980418, 0,
1.39475089522996620097633980418, 3.08138258551269292699289428468, 3.44520716762330185605204366758, 4.15944903477114194023122068929, 5.69028292856084874870147978156, 6.70631343420272650918033110902, 7.25646776305200764167941850661, 8.349946623726534446920066140609, 8.946069198867718019916370215626
