L(s) = 1 | + 0.733·2-s − 3-s − 1.46·4-s − 5-s − 0.733·6-s − 2.61·7-s − 2.53·8-s + 9-s − 0.733·10-s + 5.62·11-s + 1.46·12-s − 13-s − 1.91·14-s + 15-s + 1.06·16-s − 4.98·17-s + 0.733·18-s − 6.33·19-s + 1.46·20-s + 2.61·21-s + 4.12·22-s − 2.74·23-s + 2.53·24-s + 25-s − 0.733·26-s − 27-s + 3.82·28-s + ⋯ |
L(s) = 1 | + 0.518·2-s − 0.577·3-s − 0.731·4-s − 0.447·5-s − 0.299·6-s − 0.987·7-s − 0.897·8-s + 0.333·9-s − 0.231·10-s + 1.69·11-s + 0.422·12-s − 0.277·13-s − 0.512·14-s + 0.258·15-s + 0.266·16-s − 1.20·17-s + 0.172·18-s − 1.45·19-s + 0.327·20-s + 0.570·21-s + 0.879·22-s − 0.572·23-s + 0.518·24-s + 0.200·25-s − 0.143·26-s − 0.192·27-s + 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5986828896\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5986828896\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 0.733T + 2T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 + 6.33T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 - 8.08T + 59T^{2} \) |
| 61 | \( 1 + 9.72T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 - 2.18T + 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 6.18T + 97T^{2} \) |
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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
−8.309661745067284094332292183402, −6.97385775718319946214418705404, −6.44158680128259485895371296900, −6.19737394675461044500461007852, −4.99007031831255899098732481704, −4.40369405502173659495947354314, −3.84986346759100720163105217538, −3.13608943364351723657031328222, −1.80070517885666231691859649554, −0.38026660643540086042980123387,
0.38026660643540086042980123387, 1.80070517885666231691859649554, 3.13608943364351723657031328222, 3.84986346759100720163105217538, 4.40369405502173659495947354314, 4.99007031831255899098732481704, 6.19737394675461044500461007852, 6.44158680128259485895371296900, 6.97385775718319946214418705404, 8.309661745067284094332292183402