L(s) = 1 | + 2-s − 3-s + 4-s + 1.97·5-s − 6-s + 4.65·7-s + 8-s + 9-s + 1.97·10-s + 2.35·11-s − 12-s − 13-s + 4.65·14-s − 1.97·15-s + 16-s − 1.40·17-s + 18-s − 4.22·19-s + 1.97·20-s − 4.65·21-s + 2.35·22-s − 1.68·23-s − 24-s − 1.09·25-s − 26-s − 27-s + 4.65·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.883·5-s − 0.408·6-s + 1.75·7-s + 0.353·8-s + 0.333·9-s + 0.624·10-s + 0.710·11-s − 0.288·12-s − 0.277·13-s + 1.24·14-s − 0.510·15-s + 0.250·16-s − 0.339·17-s + 0.235·18-s − 0.969·19-s + 0.441·20-s − 1.01·21-s + 0.502·22-s − 0.352·23-s − 0.204·24-s − 0.219·25-s − 0.196·26-s − 0.192·27-s + 0.879·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.316755040\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.316755040\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 1.97T + 5T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + 4.22T + 19T^{2} \) |
| 23 | \( 1 + 1.68T + 23T^{2} \) |
| 29 | \( 1 - 0.920T + 29T^{2} \) |
| 31 | \( 1 + 4.23T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 + 8.28T + 41T^{2} \) |
| 43 | \( 1 - 9.40T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 4.69T + 59T^{2} \) |
| 61 | \( 1 + 1.74T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 6.96T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.59T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 - 3.99T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.66558256652507730742505756097, −6.99069771083060843677648069722, −6.29446445713303262559950366216, −5.54430308366469998156183834182, −5.20859329627663376284047782558, −4.28940144700682542177144904626, −3.93649482942917825793480685545, −2.30852178544222191466563083330, −1.98852152092006543887556416834, −1.00632472357163755265413174734,
1.00632472357163755265413174734, 1.98852152092006543887556416834, 2.30852178544222191466563083330, 3.93649482942917825793480685545, 4.28940144700682542177144904626, 5.20859329627663376284047782558, 5.54430308366469998156183834182, 6.29446445713303262559950366216, 6.99069771083060843677648069722, 7.66558256652507730742505756097