| L(s) = 1 | − 56.4·3-s + 2.18e3·5-s − 1.64e4·9-s − 8.54e4·11-s + 1.27e5·13-s − 1.23e5·15-s − 1.87e5·17-s + 5.51e5·19-s − 2.38e6·23-s + 2.83e6·25-s + 2.04e6·27-s + 7.39e6·29-s + 3.67e6·31-s + 4.82e6·33-s − 1.11e7·37-s − 7.19e6·39-s + 2.57e7·41-s + 4.18e6·43-s − 3.60e7·45-s − 1.27e7·47-s + 1.05e7·51-s − 4.87e6·53-s − 1.86e8·55-s − 3.11e7·57-s − 5.82e6·59-s − 3.68e6·61-s + 2.78e8·65-s + ⋯ |
| L(s) = 1 | − 0.402·3-s + 1.56·5-s − 0.838·9-s − 1.75·11-s + 1.23·13-s − 0.629·15-s − 0.544·17-s + 0.971·19-s − 1.78·23-s + 1.45·25-s + 0.739·27-s + 1.94·29-s + 0.714·31-s + 0.707·33-s − 0.978·37-s − 0.497·39-s + 1.42·41-s + 0.186·43-s − 1.31·45-s − 0.381·47-s + 0.219·51-s − 0.0849·53-s − 2.75·55-s − 0.390·57-s − 0.0625·59-s − 0.0340·61-s + 1.93·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.148695143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148695143\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 56.4T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.18e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 8.54e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.27e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.87e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.51e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.38e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.39e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.11e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.57e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.18e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.27e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.87e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.82e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.68e6T + 1.16e16T^{2} \) |
| 67 | \( 1 + 9.58e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.60e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.36e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.51e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.36e8T + 7.60e17T^{2} \) |
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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
−10.57769912175426937979073563309, −10.13263238013731977710787805288, −8.875480361901809281856019583635, −7.962350242075380259783126512002, −6.30379300490742694355874272097, −5.80172524437048734221496244928, −4.87789146860158338131115810780, −2.99506739677341213110259059926, −2.09099872790193974776395998576, −0.70877819848339789188764106605,
0.70877819848339789188764106605, 2.09099872790193974776395998576, 2.99506739677341213110259059926, 4.87789146860158338131115810780, 5.80172524437048734221496244928, 6.30379300490742694355874272097, 7.962350242075380259783126512002, 8.875480361901809281856019583635, 10.13263238013731977710787805288, 10.57769912175426937979073563309
