L(s) = 1 | − 54.1·2-s − 3.49e3·3-s − 1.34e4·4-s − 1.04e5i·5-s + 1.89e5·6-s − 3.63e5i·7-s + 1.61e6·8-s + 7.44e6·9-s + 5.63e6i·10-s + 1.70e7i·11-s + 4.70e7·12-s + 4.63e7·13-s + 1.96e7i·14-s + 3.63e8i·15-s + 1.32e8·16-s − 7.43e8i·17-s + ⋯ |
L(s) = 1 | − 0.422·2-s − 1.59·3-s − 0.821·4-s − 1.33i·5-s + 0.676·6-s − 0.441i·7-s + 0.770·8-s + 1.55·9-s + 0.563i·10-s + 0.874i·11-s + 1.31·12-s + 0.739·13-s + 0.186i·14-s + 2.12i·15-s + 0.495·16-s − 1.81i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.459i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(0.6091660671\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6091660671\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (-3.02e9 + 1.56e9i)T \) |
good | 2 | \( 1 + 54.1T + 1.63e4T^{2} \) |
| 3 | \( 1 + 3.49e3T + 4.78e6T^{2} \) |
| 5 | \( 1 + 1.04e5iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 3.63e5iT - 6.78e11T^{2} \) |
| 11 | \( 1 - 1.70e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 - 4.63e7T + 3.93e15T^{2} \) |
| 17 | \( 1 + 7.43e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 1.19e9iT - 7.99e17T^{2} \) |
| 29 | \( 1 - 1.91e10T + 2.97e20T^{2} \) |
| 31 | \( 1 - 1.15e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + 4.82e10iT - 9.01e21T^{2} \) |
| 41 | \( 1 + 2.83e11T + 3.79e22T^{2} \) |
| 43 | \( 1 - 2.89e11iT - 7.38e22T^{2} \) |
| 47 | \( 1 - 1.70e11T + 2.56e23T^{2} \) |
| 53 | \( 1 + 2.00e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 - 1.84e12T + 6.19e24T^{2} \) |
| 61 | \( 1 - 2.15e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 + 3.48e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 - 5.84e12T + 8.27e25T^{2} \) |
| 73 | \( 1 - 1.12e13T + 1.22e26T^{2} \) |
| 79 | \( 1 - 1.67e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 + 2.84e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 7.66e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 1.08e14iT - 6.52e27T^{2} \) |
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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
−13.55635412049531492072359615487, −12.62457245865707515253343493891, −11.40856090319413414009144556276, −9.984541944960594002385583951049, −8.802612254347636203865747608344, −6.98621279781411292117224565659, −5.02533610098185915857299012164, −4.66377664192476455048459268829, −0.954684570682528670698973901448, −0.47008219179423057328614104671,
1.15452255852852877388374975050, 3.71297589445307809845168705391, 5.57088942264483218467985454533, 6.50138373369736043763899678834, 8.351398605006785946560315829666, 10.32243058813302337600698075193, 10.85372331092817188145719902171, 12.24666100352616097224705602990, 13.75229159248869278795466544021, 15.22739724674735334815753077538