| L(s) = 1 | + 2-s − 3·3-s − 7·4-s − 12·5-s − 3·6-s − 20·7-s − 15·8-s + 9·9-s − 12·10-s − 4·11-s + 21·12-s + 76·13-s − 20·14-s + 36·15-s + 41·16-s + 22·17-s + 9·18-s + 84·20-s + 60·21-s − 4·22-s + 82·23-s + 45·24-s + 19·25-s + 76·26-s − 27·27-s + 140·28-s − 242·29-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s − 1.07·5-s − 0.204·6-s − 1.07·7-s − 0.662·8-s + 1/3·9-s − 0.379·10-s − 0.109·11-s + 0.505·12-s + 1.62·13-s − 0.381·14-s + 0.619·15-s + 0.640·16-s + 0.313·17-s + 0.117·18-s + 0.939·20-s + 0.623·21-s − 0.0387·22-s + 0.743·23-s + 0.382·24-s + 0.151·25-s + 0.573·26-s − 0.192·27-s + 0.944·28-s − 1.54·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 76 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 23 | \( 1 - 82 T + p^{3} T^{2} \) |
| 29 | \( 1 + 242 T + p^{3} T^{2} \) |
| 31 | \( 1 - 126 T + p^{3} T^{2} \) |
| 37 | \( 1 - 180 T + p^{3} T^{2} \) |
| 41 | \( 1 - 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 308 T + p^{3} T^{2} \) |
| 47 | \( 1 + 522 T + p^{3} T^{2} \) |
| 53 | \( 1 - 70 T + p^{3} T^{2} \) |
| 59 | \( 1 + 188 T + p^{3} T^{2} \) |
| 61 | \( 1 + 706 T + p^{3} T^{2} \) |
| 67 | \( 1 + 104 T + p^{3} T^{2} \) |
| 71 | \( 1 - 432 T + p^{3} T^{2} \) |
| 73 | \( 1 - 718 T + p^{3} T^{2} \) |
| 79 | \( 1 + 94 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1296 T + p^{3} T^{2} \) |
| 89 | \( 1 + 846 T + p^{3} T^{2} \) |
| 97 | \( 1 + 830 T + p^{3} T^{2} \) |
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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
−9.160271589402270976408109520017, −8.249962788334443005741485354870, −7.44081481822011889091134399463, −6.26919706364655544176094888769, −5.75525767772265521607407717880, −4.54961899759875406004252006029, −3.80244219032152589562823617417, −3.16252768986189697423571496283, −0.972523399241738064682902762509, 0,
0.972523399241738064682902762509, 3.16252768986189697423571496283, 3.80244219032152589562823617417, 4.54961899759875406004252006029, 5.75525767772265521607407717880, 6.26919706364655544176094888769, 7.44081481822011889091134399463, 8.249962788334443005741485354870, 9.160271589402270976408109520017
