L(s) = 1 | + 2.10·2-s + 3.19·3-s + 2.41·4-s + 5-s + 6.70·6-s − 3.48·7-s + 0.872·8-s + 7.19·9-s + 2.10·10-s + 2.99·11-s + 7.71·12-s + 2.47·13-s − 7.33·14-s + 3.19·15-s − 2.99·16-s + 1.89·17-s + 15.1·18-s − 0.418·19-s + 2.41·20-s − 11.1·21-s + 6.28·22-s + 2.78·24-s + 25-s + 5.20·26-s + 13.3·27-s − 8.42·28-s − 7.10·29-s + ⋯ |
L(s) = 1 | + 1.48·2-s + 1.84·3-s + 1.20·4-s + 0.447·5-s + 2.73·6-s − 1.31·7-s + 0.308·8-s + 2.39·9-s + 0.664·10-s + 0.901·11-s + 2.22·12-s + 0.687·13-s − 1.95·14-s + 0.824·15-s − 0.749·16-s + 0.460·17-s + 3.56·18-s − 0.0959·19-s + 0.540·20-s − 2.43·21-s + 1.33·22-s + 0.568·24-s + 0.200·25-s + 1.02·26-s + 2.57·27-s − 1.59·28-s − 1.31·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.832865884\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.832865884\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 3 | \( 1 - 3.19T + 3T^{2} \) |
| 7 | \( 1 + 3.48T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 + 0.418T + 19T^{2} \) |
| 29 | \( 1 + 7.10T + 29T^{2} \) |
| 31 | \( 1 - 4.36T + 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 - 7.26T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 3.80T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 4.30T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 - 8.64T + 73T^{2} \) |
| 79 | \( 1 + 7.77T + 79T^{2} \) |
| 83 | \( 1 - 7.60T + 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 - 3.81T + 97T^{2} \) |
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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
−8.995053125287204995846521165712, −8.131304559992227306243137642901, −7.08164561565859504152058665960, −6.51487366535492241289312299335, −5.81682584628136783230374300023, −4.60049229904977092243681887360, −3.70136316998137540672161433448, −3.40664781910563335410857922609, −2.63412851880317028110430801466, −1.61568179566997135440973055518,
1.61568179566997135440973055518, 2.63412851880317028110430801466, 3.40664781910563335410857922609, 3.70136316998137540672161433448, 4.60049229904977092243681887360, 5.81682584628136783230374300023, 6.51487366535492241289312299335, 7.08164561565859504152058665960, 8.131304559992227306243137642901, 8.995053125287204995846521165712