| L(s) = 1 | − 2.51·2-s + 2.36·3-s + 4.31·4-s + 5-s − 5.95·6-s − 5.81·8-s + 2.61·9-s − 2.51·10-s − 1.11·11-s + 10.2·12-s − 4.71·13-s + 2.36·15-s + 5.98·16-s − 3.73·17-s − 6.56·18-s + 4.50·19-s + 4.31·20-s + 2.80·22-s + 3.90·23-s − 13.7·24-s + 25-s + 11.8·26-s − 0.920·27-s − 0.0568·29-s − 5.95·30-s − 31-s − 3.40·32-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 1.36·3-s + 2.15·4-s + 0.447·5-s − 2.43·6-s − 2.05·8-s + 0.870·9-s − 0.794·10-s − 0.336·11-s + 2.94·12-s − 1.30·13-s + 0.611·15-s + 1.49·16-s − 0.905·17-s − 1.54·18-s + 1.03·19-s + 0.964·20-s + 0.597·22-s + 0.814·23-s − 2.81·24-s + 0.200·25-s + 2.32·26-s − 0.177·27-s − 0.0105·29-s − 1.08·30-s − 0.179·31-s − 0.601·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 - 2.36T + 3T^{2} \) |
| 11 | \( 1 + 1.11T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 + 0.0568T + 29T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 4.65T + 43T^{2} \) |
| 47 | \( 1 + 3.56T + 47T^{2} \) |
| 53 | \( 1 + 0.737T + 53T^{2} \) |
| 59 | \( 1 + 4.02T + 59T^{2} \) |
| 61 | \( 1 - 8.46T + 61T^{2} \) |
| 67 | \( 1 - 0.150T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 0.210T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 9.25T + 83T^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 - 2.56T + 97T^{2} \) |
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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
−7.72714973901346686206816416397, −7.15836986641455941499504393313, −6.73340557144464786429350044084, −5.56028908629195723370906826379, −4.71365344783598287427289393776, −3.42880037309449303409475329732, −2.65569712517842182659570871996, −2.21713552347545748141292197443, −1.33821560990711895950696949959, 0,
1.33821560990711895950696949959, 2.21713552347545748141292197443, 2.65569712517842182659570871996, 3.42880037309449303409475329732, 4.71365344783598287427289393776, 5.56028908629195723370906826379, 6.73340557144464786429350044084, 7.15836986641455941499504393313, 7.72714973901346686206816416397
