| L(s) = 1 | − 6·3-s + 6·5-s + 9·9-s − 4·11-s + 34·13-s − 36·15-s + 54·17-s − 144·19-s + 8·23-s − 89·25-s + 108·27-s + 180·29-s − 128·31-s + 24·33-s − 232·37-s − 204·39-s + 446·41-s + 86·43-s + 54·45-s + 192·47-s − 343·49-s − 324·51-s − 348·53-s − 24·55-s + 864·57-s + 838·59-s + 780·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.536·5-s + 1/3·9-s − 0.109·11-s + 0.725·13-s − 0.619·15-s + 0.770·17-s − 1.73·19-s + 0.0725·23-s − 0.711·25-s + 0.769·27-s + 1.15·29-s − 0.741·31-s + 0.126·33-s − 1.03·37-s − 0.837·39-s + 1.69·41-s + 0.304·43-s + 0.178·45-s + 0.595·47-s − 49-s − 0.889·51-s − 0.901·53-s − 0.0588·55-s + 2.00·57-s + 1.84·59-s + 1.63·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 79 | \( 1 + p T \) |
| good | 3 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 144 T + p^{3} T^{2} \) |
| 23 | \( 1 - 8 T + p^{3} T^{2} \) |
| 29 | \( 1 - 180 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 232 T + p^{3} T^{2} \) |
| 41 | \( 1 - 446 T + p^{3} T^{2} \) |
| 43 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 192 T + p^{3} T^{2} \) |
| 53 | \( 1 + 348 T + p^{3} T^{2} \) |
| 59 | \( 1 - 838 T + p^{3} T^{2} \) |
| 61 | \( 1 - 780 T + p^{3} T^{2} \) |
| 67 | \( 1 + 904 T + p^{3} T^{2} \) |
| 71 | \( 1 + 312 T + p^{3} T^{2} \) |
| 73 | \( 1 + 234 T + p^{3} T^{2} \) |
| 83 | \( 1 - 316 T + p^{3} T^{2} \) |
| 89 | \( 1 - 566 T + p^{3} T^{2} \) |
| 97 | \( 1 + 806 T + p^{3} T^{2} \) |
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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
−8.893041228149239255566071168708, −8.167449478690642719855097099475, −7.00780912114368212916650357937, −6.15238067788157557722971755043, −5.75319295022958945563702058419, −4.80070924785490978564808296540, −3.79540093576808109965865774987, −2.42618723366247302759454375901, −1.20046435204998555758718939225, 0,
1.20046435204998555758718939225, 2.42618723366247302759454375901, 3.79540093576808109965865774987, 4.80070924785490978564808296540, 5.75319295022958945563702058419, 6.15238067788157557722971755043, 7.00780912114368212916650357937, 8.167449478690642719855097099475, 8.893041228149239255566071168708
