L(s) = 1 | − 2-s − 3-s + 4-s + 3.82·5-s + 6-s − 0.790·7-s − 8-s + 9-s − 3.82·10-s + 0.446·11-s − 12-s + 0.193·13-s + 0.790·14-s − 3.82·15-s + 16-s − 4.41·17-s − 18-s − 19-s + 3.82·20-s + 0.790·21-s − 0.446·22-s + 4.46·23-s + 24-s + 9.66·25-s − 0.193·26-s − 27-s − 0.790·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.71·5-s + 0.408·6-s − 0.298·7-s − 0.353·8-s + 0.333·9-s − 1.21·10-s + 0.134·11-s − 0.288·12-s + 0.0537·13-s + 0.211·14-s − 0.988·15-s + 0.250·16-s − 1.07·17-s − 0.235·18-s − 0.229·19-s + 0.856·20-s + 0.172·21-s − 0.0952·22-s + 0.930·23-s + 0.204·24-s + 1.93·25-s − 0.0380·26-s − 0.192·27-s − 0.149·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.521689281\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.521689281\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 + 0.790T + 7T^{2} \) |
| 11 | \( 1 - 0.446T + 11T^{2} \) |
| 13 | \( 1 - 0.193T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 + 3.07T + 29T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 + 0.0299T + 41T^{2} \) |
| 43 | \( 1 - 0.658T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 - 8.27T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 8.46T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.236321266227850822928989233322, −7.08342997026104198925895704133, −6.66543302552573781586425312690, −6.08903335567004197917115379189, −5.39027140496462015557506968774, −4.71974161632054434932931490871, −3.47436754746646012596228035910, −2.38694264051638749270700240973, −1.81739584912946824172333081621, −0.74587008656107827281545288313,
0.74587008656107827281545288313, 1.81739584912946824172333081621, 2.38694264051638749270700240973, 3.47436754746646012596228035910, 4.71974161632054434932931490871, 5.39027140496462015557506968774, 6.08903335567004197917115379189, 6.66543302552573781586425312690, 7.08342997026104198925895704133, 8.236321266227850822928989233322