L(s) = 1 | − 3.57·2-s − 3·3-s + 4.79·4-s + 5·5-s + 10.7·6-s + 15.0·7-s + 11.4·8-s + 9·9-s − 17.8·10-s + 70.8·11-s − 14.3·12-s + 62.2·13-s − 53.7·14-s − 15·15-s − 79.3·16-s − 67.1·17-s − 32.1·18-s + 118.·19-s + 23.9·20-s − 45.0·21-s − 253.·22-s + 72.5·23-s − 34.3·24-s + 25·25-s − 222.·26-s − 27·27-s + 72.0·28-s + ⋯ |
L(s) = 1 | − 1.26·2-s − 0.577·3-s + 0.599·4-s + 0.447·5-s + 0.730·6-s + 0.811·7-s + 0.506·8-s + 0.333·9-s − 0.565·10-s + 1.94·11-s − 0.346·12-s + 1.32·13-s − 1.02·14-s − 0.258·15-s − 1.24·16-s − 0.958·17-s − 0.421·18-s + 1.43·19-s + 0.268·20-s − 0.468·21-s − 2.45·22-s + 0.657·23-s − 0.292·24-s + 0.200·25-s − 1.67·26-s − 0.192·27-s + 0.486·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.192949526\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.192949526\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
good | 2 | \( 1 + 3.57T + 8T^{2} \) |
| 7 | \( 1 - 15.0T + 343T^{2} \) |
| 11 | \( 1 - 70.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 72.5T + 1.21e4T^{2} \) |
| 31 | \( 1 + 180.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 47.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 409.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 125.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 215.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 356.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 578.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 870.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 411.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 388.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.52e3T + 9.12e5T^{2} \) |
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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
−10.92505369369704229975299595250, −9.492825604155745370908771682853, −9.142342241576720618126581583283, −8.207811901967754044887409949826, −7.05846458998817516806568281796, −6.30348231940750811554434594808, −5.01200174110809291543161844425, −3.84962968502172973922307146854, −1.64845617539078381701702313123, −1.00545073043389954214883431763,
1.00545073043389954214883431763, 1.64845617539078381701702313123, 3.84962968502172973922307146854, 5.01200174110809291543161844425, 6.30348231940750811554434594808, 7.05846458998817516806568281796, 8.207811901967754044887409949826, 9.142342241576720618126581583283, 9.492825604155745370908771682853, 10.92505369369704229975299595250