L(s) = 1 | − 1.56·2-s + 3-s + 0.438·4-s − 3.56·5-s − 1.56·6-s + 0.561·7-s + 2.43·8-s + 9-s + 5.56·10-s − 2·11-s + 0.438·12-s − 0.876·14-s − 3.56·15-s − 4.68·16-s − 1.56·17-s − 1.56·18-s + 7.12·19-s − 1.56·20-s + 0.561·21-s + 3.12·22-s + 2·23-s + 2.43·24-s + 7.68·25-s + 27-s + 0.246·28-s + 6.68·29-s + 5.56·30-s + ⋯ |
L(s) = 1 | − 1.10·2-s + 0.577·3-s + 0.219·4-s − 1.59·5-s − 0.637·6-s + 0.212·7-s + 0.862·8-s + 0.333·9-s + 1.75·10-s − 0.603·11-s + 0.126·12-s − 0.234·14-s − 0.919·15-s − 1.17·16-s − 0.378·17-s − 0.368·18-s + 1.63·19-s − 0.349·20-s + 0.122·21-s + 0.665·22-s + 0.417·23-s + 0.497·24-s + 1.53·25-s + 0.192·27-s + 0.0465·28-s + 1.24·29-s + 1.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6754648700\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6754648700\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 - 0.561T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 - 7.56T + 37T^{2} \) |
| 41 | \( 1 + 1.56T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 0.684T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 - 4.56T + 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 + 0.876T + 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−10.82589628888930509222156347347, −9.849230605811608010124469113557, −8.996913759646670549518095881824, −8.091320633891752412812506182646, −7.77503323002833520594102119833, −6.92326777771850493959671438728, −4.98743209888744746823088102972, −4.08495378148245155731356427379, −2.83432581445068102501033746268, −0.877085449946524228231774992588,
0.877085449946524228231774992588, 2.83432581445068102501033746268, 4.08495378148245155731356427379, 4.98743209888744746823088102972, 6.92326777771850493959671438728, 7.77503323002833520594102119833, 8.091320633891752412812506182646, 8.996913759646670549518095881824, 9.849230605811608010124469113557, 10.82589628888930509222156347347