L(s) = 1 | − 2-s + 1.60·3-s + 4-s + 5-s − 1.60·6-s − 3.11·7-s − 8-s − 0.425·9-s − 10-s − 0.687·11-s + 1.60·12-s + 0.929·13-s + 3.11·14-s + 1.60·15-s + 16-s − 2.78·17-s + 0.425·18-s − 2.69·19-s + 20-s − 5.00·21-s + 0.687·22-s − 7.77·23-s − 1.60·24-s + 25-s − 0.929·26-s − 5.49·27-s − 3.11·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.926·3-s + 0.5·4-s + 0.447·5-s − 0.655·6-s − 1.17·7-s − 0.353·8-s − 0.141·9-s − 0.316·10-s − 0.207·11-s + 0.463·12-s + 0.257·13-s + 0.832·14-s + 0.414·15-s + 0.250·16-s − 0.674·17-s + 0.100·18-s − 0.617·19-s + 0.223·20-s − 1.09·21-s + 0.146·22-s − 1.62·23-s − 0.327·24-s + 0.200·25-s − 0.182·26-s − 1.05·27-s − 0.588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.403630063\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.403630063\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 1.60T + 3T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 + 0.687T + 11T^{2} \) |
| 13 | \( 1 - 0.929T + 13T^{2} \) |
| 17 | \( 1 + 2.78T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 - 8.25T + 31T^{2} \) |
| 37 | \( 1 + 2.56T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 6.09T + 43T^{2} \) |
| 47 | \( 1 - 8.18T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 5.82T + 67T^{2} \) |
| 71 | \( 1 - 3.66T + 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 2.01T + 89T^{2} \) |
| 97 | \( 1 + 0.940T + 97T^{2} \) |
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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
−8.322646477453384409728277208018, −7.57732782133288911245787606564, −6.52581683306513104038337943756, −6.33499013865408354237630291349, −5.40918803616172041724019366016, −4.14946690232952102110076284393, −3.45869309899544477246842206215, −2.49289450873795498462703992674, −2.17594143985927227831713766569, −0.62383805089280419656575166310,
0.62383805089280419656575166310, 2.17594143985927227831713766569, 2.49289450873795498462703992674, 3.45869309899544477246842206215, 4.14946690232952102110076284393, 5.40918803616172041724019366016, 6.33499013865408354237630291349, 6.52581683306513104038337943756, 7.57732782133288911245787606564, 8.322646477453384409728277208018