L(s) = 1 | + 2.68·2-s − 1.91·3-s + 5.19·4-s + 5-s − 5.14·6-s − 7-s + 8.56·8-s + 0.681·9-s + 2.68·10-s − 3.48·11-s − 9.96·12-s − 2.77·13-s − 2.68·14-s − 1.91·15-s + 12.5·16-s − 2.05·17-s + 1.82·18-s − 1.06·19-s + 5.19·20-s + 1.91·21-s − 9.34·22-s − 0.510·23-s − 16.4·24-s + 25-s − 7.43·26-s + 4.44·27-s − 5.19·28-s + ⋯ |
L(s) = 1 | + 1.89·2-s − 1.10·3-s + 2.59·4-s + 0.447·5-s − 2.10·6-s − 0.377·7-s + 3.02·8-s + 0.227·9-s + 0.848·10-s − 1.05·11-s − 2.87·12-s − 0.769·13-s − 0.716·14-s − 0.495·15-s + 3.14·16-s − 0.498·17-s + 0.430·18-s − 0.244·19-s + 1.16·20-s + 0.418·21-s − 1.99·22-s − 0.106·23-s − 3.35·24-s + 0.200·25-s − 1.45·26-s + 0.856·27-s − 0.981·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 + 2.05T + 17T^{2} \) |
| 19 | \( 1 + 1.06T + 19T^{2} \) |
| 23 | \( 1 + 0.510T + 23T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 5.32T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 3.00T + 43T^{2} \) |
| 47 | \( 1 + 1.20T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 + 3.53T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 4.21T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 8.49T + 83T^{2} \) |
| 89 | \( 1 + 3.80T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−6.95703031946700329907553636801, −6.51735819076494457562970081223, −5.90656246869860971409994277438, −5.28866539238725026992944723566, −4.87244076204716384030488688059, −4.23862406849925132457767251193, −3.05344774555238264570603601323, −2.64955442319686304962890437407, −1.62378438347236501871769771348, 0,
1.62378438347236501871769771348, 2.64955442319686304962890437407, 3.05344774555238264570603601323, 4.23862406849925132457767251193, 4.87244076204716384030488688059, 5.28866539238725026992944723566, 5.90656246869860971409994277438, 6.51735819076494457562970081223, 6.95703031946700329907553636801