| L(s) = 1 | − 0.632·2-s + 1.91·3-s − 1.59·4-s + 5-s − 1.21·6-s + 4.08·7-s + 2.27·8-s + 0.685·9-s − 0.632·10-s − 4.34·11-s − 3.07·12-s + 1.55·13-s − 2.58·14-s + 1.91·15-s + 1.75·16-s + 6.36·17-s − 0.433·18-s − 1.59·20-s + 7.84·21-s + 2.75·22-s − 3.36·23-s + 4.37·24-s + 25-s − 0.981·26-s − 4.44·27-s − 6.54·28-s + 5.21·29-s + ⋯ |
| L(s) = 1 | − 0.447·2-s + 1.10·3-s − 0.799·4-s + 0.447·5-s − 0.495·6-s + 1.54·7-s + 0.805·8-s + 0.228·9-s − 0.200·10-s − 1.31·11-s − 0.886·12-s + 0.430·13-s − 0.691·14-s + 0.495·15-s + 0.439·16-s + 1.54·17-s − 0.102·18-s − 0.357·20-s + 1.71·21-s + 0.586·22-s − 0.701·23-s + 0.892·24-s + 0.200·25-s − 0.192·26-s − 0.855·27-s − 1.23·28-s + 0.967·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.134715894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.134715894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.632T + 2T^{2} \) |
| 3 | \( 1 - 1.91T + 3T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 - 6.36T + 17T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 0.180T + 37T^{2} \) |
| 41 | \( 1 - 0.0257T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 - 9.54T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 + 0.507T + 89T^{2} \) |
| 97 | \( 1 - 3.28T + 97T^{2} \) |
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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
−9.073290331937672894464443534297, −8.301109693966407188208415700736, −8.063114066629258820703948382256, −7.49808077887665397790886265963, −5.83302111194120592590576177971, −5.13347484123774014479918831351, −4.34771552465986071148056855808, −3.20452783616271510984941270593, −2.19144947911292563907688144199, −1.10827181142028850684445932732,
1.10827181142028850684445932732, 2.19144947911292563907688144199, 3.20452783616271510984941270593, 4.34771552465986071148056855808, 5.13347484123774014479918831351, 5.83302111194120592590576177971, 7.49808077887665397790886265963, 8.063114066629258820703948382256, 8.301109693966407188208415700736, 9.073290331937672894464443534297
