L(s) = 1 | − 2.44·2-s + 3.99·4-s + 5-s − 7-s − 4.89·8-s − 2.44·10-s + 2.44·11-s − 0.449·13-s + 2.44·14-s + 3.99·16-s + 0.550·17-s − 0.449·19-s + 3.99·20-s − 5.99·22-s + 23-s + 25-s + 1.10·26-s − 3.99·28-s + 4.34·29-s + 9.89·31-s − 1.34·34-s − 35-s − 5.89·37-s + 1.10·38-s − 4.89·40-s − 0.550·41-s + 2·43-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 1.99·4-s + 0.447·5-s − 0.377·7-s − 1.73·8-s − 0.774·10-s + 0.738·11-s − 0.124·13-s + 0.654·14-s + 0.999·16-s + 0.133·17-s − 0.103·19-s + 0.894·20-s − 1.27·22-s + 0.208·23-s + 0.200·25-s + 0.215·26-s − 0.755·28-s + 0.807·29-s + 1.77·31-s − 0.231·34-s − 0.169·35-s − 0.969·37-s + 0.178·38-s − 0.774·40-s − 0.0859·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7510636158\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7510636158\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 - 0.550T + 17T^{2} \) |
| 19 | \( 1 + 0.449T + 19T^{2} \) |
| 29 | \( 1 - 4.34T + 29T^{2} \) |
| 31 | \( 1 - 9.89T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 + 0.550T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 + 5.44T + 53T^{2} \) |
| 59 | \( 1 - 4.34T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 9.34T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−10.04208356181078830870213166871, −9.054712595090912563085757480082, −8.538659652094765271947710581392, −7.62233718444430191573123124251, −6.67731781997491447287010280237, −6.21774856444065041237593704809, −4.77009662969871197925817999419, −3.22685565937445748552169871403, −2.07150504732527634756853433338, −0.883812992174290884525133007365,
0.883812992174290884525133007365, 2.07150504732527634756853433338, 3.22685565937445748552169871403, 4.77009662969871197925817999419, 6.21774856444065041237593704809, 6.67731781997491447287010280237, 7.62233718444430191573123124251, 8.538659652094765271947710581392, 9.054712595090912563085757480082, 10.04208356181078830870213166871