L(s) = 1 | + 0.763·2-s + 0.284·3-s − 1.41·4-s + 5-s + 0.217·6-s − 2.19·7-s − 2.60·8-s − 2.91·9-s + 0.763·10-s + 11-s − 0.403·12-s − 0.847·13-s − 1.67·14-s + 0.284·15-s + 0.842·16-s − 3.79·17-s − 2.22·18-s − 3.84·19-s − 1.41·20-s − 0.625·21-s + 0.763·22-s + 5.55·23-s − 0.742·24-s + 25-s − 0.646·26-s − 1.68·27-s + 3.11·28-s + ⋯ |
L(s) = 1 | + 0.539·2-s + 0.164·3-s − 0.708·4-s + 0.447·5-s + 0.0887·6-s − 0.830·7-s − 0.922·8-s − 0.973·9-s + 0.241·10-s + 0.301·11-s − 0.116·12-s − 0.234·13-s − 0.448·14-s + 0.0734·15-s + 0.210·16-s − 0.920·17-s − 0.525·18-s − 0.882·19-s − 0.316·20-s − 0.136·21-s + 0.162·22-s + 1.15·23-s − 0.151·24-s + 0.200·25-s − 0.126·26-s − 0.324·27-s + 0.588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.400822951\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.400822951\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 - 0.763T + 2T^{2} \) |
| 3 | \( 1 - 0.284T + 3T^{2} \) |
| 7 | \( 1 + 2.19T + 7T^{2} \) |
| 13 | \( 1 + 0.847T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 3.84T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 - 8.71T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 5.30T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 9.18T + 59T^{2} \) |
| 61 | \( 1 - 3.85T + 61T^{2} \) |
| 67 | \( 1 + 6.03T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 2.31T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.759944265084888835100228787043, −7.73342977394824467657178151504, −6.76825739450861930855699102517, −6.03040918158669551065964873816, −5.58671834137010184868081732211, −4.58964905824766738332208620257, −3.92111197422178828143998745042, −2.98486522945995560746635773974, −2.33317952892054101086360741836, −0.59424519883884972212965280051,
0.59424519883884972212965280051, 2.33317952892054101086360741836, 2.98486522945995560746635773974, 3.92111197422178828143998745042, 4.58964905824766738332208620257, 5.58671834137010184868081732211, 6.03040918158669551065964873816, 6.76825739450861930855699102517, 7.73342977394824467657178151504, 8.759944265084888835100228787043