L(s) = 1 | − 3-s − 4.11·5-s + 9-s − 5.49·11-s + 3.13·13-s + 4.11·15-s + 17-s − 3.29·19-s − 5.91·23-s + 11.9·25-s − 27-s − 7.47·29-s + 1.10·31-s + 5.49·33-s − 5.90·37-s − 3.13·39-s + 1.51·41-s − 12.2·43-s − 4.11·45-s − 1.40·47-s − 51-s − 11.8·53-s + 22.6·55-s + 3.29·57-s + 14.3·59-s − 11.1·61-s − 12.8·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.84·5-s + 0.333·9-s − 1.65·11-s + 0.869·13-s + 1.06·15-s + 0.242·17-s − 0.757·19-s − 1.23·23-s + 2.38·25-s − 0.192·27-s − 1.38·29-s + 0.198·31-s + 0.956·33-s − 0.970·37-s − 0.501·39-s + 0.237·41-s − 1.86·43-s − 0.613·45-s − 0.204·47-s − 0.140·51-s − 1.62·53-s + 3.04·55-s + 0.437·57-s + 1.86·59-s − 1.42·61-s − 1.59·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9996 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9996 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06325047772\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06325047772\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 4.11T + 5T^{2} \) |
| 11 | \( 1 + 5.49T + 11T^{2} \) |
| 13 | \( 1 - 3.13T + 13T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 - 1.51T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 4.33T + 73T^{2} \) |
| 79 | \( 1 + 4.77T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 + 4.11T + 97T^{2} \) |
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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
−7.79156667805054160863759940537, −7.05114949217887035799935236303, −6.38192716746703541456330650685, −5.46061803362512091933509726370, −4.95264504659661999843371631712, −4.04945821641826065771312152523, −3.65305271852278815611612699255, −2.75217514815349566884557630684, −1.56420080972939221288974659435, −0.12171024029698534207647921609,
0.12171024029698534207647921609, 1.56420080972939221288974659435, 2.75217514815349566884557630684, 3.65305271852278815611612699255, 4.04945821641826065771312152523, 4.95264504659661999843371631712, 5.46061803362512091933509726370, 6.38192716746703541456330650685, 7.05114949217887035799935236303, 7.79156667805054160863759940537