| L(s) = 1 | − 2.38·2-s − 0.126·3-s + 3.67·4-s + 5-s + 0.300·6-s − 4.61·7-s − 3.99·8-s − 2.98·9-s − 2.38·10-s − 0.463·12-s + 13-s + 11.0·14-s − 0.126·15-s + 2.16·16-s + 0.706·17-s + 7.10·18-s − 7.92·19-s + 3.67·20-s + 0.582·21-s + 2.17·23-s + 0.503·24-s + 25-s − 2.38·26-s + 0.755·27-s − 16.9·28-s − 1.60·29-s + 0.300·30-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.0728·3-s + 1.83·4-s + 0.447·5-s + 0.122·6-s − 1.74·7-s − 1.41·8-s − 0.994·9-s − 0.753·10-s − 0.133·12-s + 0.277·13-s + 2.94·14-s − 0.0325·15-s + 0.540·16-s + 0.171·17-s + 1.67·18-s − 1.81·19-s + 0.821·20-s + 0.127·21-s + 0.454·23-s + 0.102·24-s + 0.200·25-s − 0.467·26-s + 0.145·27-s − 3.20·28-s − 0.297·29-s + 0.0548·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 + 0.126T + 3T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 17 | \( 1 - 0.706T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 - 7.64T + 31T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 - 0.274T + 41T^{2} \) |
| 43 | \( 1 - 9.87T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 3.85T + 59T^{2} \) |
| 61 | \( 1 + 7.25T + 61T^{2} \) |
| 67 | \( 1 - 9.78T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 - 9.03T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.75692782789772933129672538350, −6.69337070656973508731690403471, −6.35505333691354822360903393507, −5.99172412396286829428371393697, −4.73777103361769126741397998963, −3.55656010570698132061826081607, −2.75169749456807159384288596202, −2.18086869412022981999707548828, −0.852671510587556166295088582439, 0,
0.852671510587556166295088582439, 2.18086869412022981999707548828, 2.75169749456807159384288596202, 3.55656010570698132061826081607, 4.73777103361769126741397998963, 5.99172412396286829428371393697, 6.35505333691354822360903393507, 6.69337070656973508731690403471, 7.75692782789772933129672538350
