| L(s) = 1 | + 9.15·3-s − 5·5-s + 11.4·7-s + 56.7·9-s − 28.6·11-s − 66.8·13-s − 45.7·15-s − 78.9·17-s + 19·19-s + 104.·21-s − 181.·23-s + 25·25-s + 272.·27-s + 259.·29-s − 335.·31-s − 261.·33-s − 57.2·35-s + 157.·37-s − 611.·39-s − 125.·41-s − 138.·43-s − 283.·45-s − 144.·47-s − 211.·49-s − 722.·51-s − 243.·53-s + 143.·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s − 0.447·5-s + 0.618·7-s + 2.10·9-s − 0.784·11-s − 1.42·13-s − 0.787·15-s − 1.12·17-s + 0.229·19-s + 1.08·21-s − 1.64·23-s + 0.200·25-s + 1.94·27-s + 1.65·29-s − 1.94·31-s − 1.38·33-s − 0.276·35-s + 0.699·37-s − 2.51·39-s − 0.477·41-s − 0.490·43-s − 0.940·45-s − 0.448·47-s − 0.617·49-s − 1.98·51-s − 0.630·53-s + 0.350·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 9.15T + 27T^{2} \) |
| 7 | \( 1 - 11.4T + 343T^{2} \) |
| 11 | \( 1 + 28.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.9T + 4.91e3T^{2} \) |
| 23 | \( 1 + 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 335.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 157.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 144.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 243.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 49.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 831.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 524.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 710.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 406.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 961.T + 9.12e5T^{2} \) |
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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
−8.592318775077954336335947280086, −7.82174981375930345517728446508, −7.61273689611264305581759012272, −6.54769540105088417779137758121, −4.99329244679100909354069615498, −4.39751173849159919147015774484, −3.37827654730149764944457448483, −2.45860003937025062400608382910, −1.83001943954149577299391032431, 0,
1.83001943954149577299391032431, 2.45860003937025062400608382910, 3.37827654730149764944457448483, 4.39751173849159919147015774484, 4.99329244679100909354069615498, 6.54769540105088417779137758121, 7.61273689611264305581759012272, 7.82174981375930345517728446508, 8.592318775077954336335947280086
