| L(s) = 1 | + 10.9·2-s + 23.6·3-s + 88.1·4-s − 25·5-s + 259.·6-s + 615.·8-s + 316.·9-s − 274.·10-s − 92.0·11-s + 2.08e3·12-s − 166.·13-s − 591.·15-s + 3.92e3·16-s + 84.6·17-s + 3.46e3·18-s − 1.33e3·19-s − 2.20e3·20-s − 1.00e3·22-s + 1.17e3·23-s + 1.45e4·24-s + 625·25-s − 1.82e3·26-s + 1.72e3·27-s − 5.10e3·29-s − 6.48e3·30-s − 7.55e3·31-s + 2.33e4·32-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 1.51·3-s + 2.75·4-s − 0.447·5-s + 2.93·6-s + 3.40·8-s + 1.30·9-s − 0.866·10-s − 0.229·11-s + 4.17·12-s − 0.272·13-s − 0.678·15-s + 3.83·16-s + 0.0710·17-s + 2.52·18-s − 0.847·19-s − 1.23·20-s − 0.444·22-s + 0.461·23-s + 5.16·24-s + 0.200·25-s − 0.528·26-s + 0.456·27-s − 1.12·29-s − 1.31·30-s − 1.41·31-s + 4.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(10.80652046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.80652046\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 10.9T + 32T^{2} \) |
| 3 | \( 1 - 23.6T + 243T^{2} \) |
| 11 | \( 1 + 92.0T + 1.61e5T^{2} \) |
| 13 | \( 1 + 166.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 84.6T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.10e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.54e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.65e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.85e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.13e5T + 8.58e9T^{2} \) |
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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
−11.50886154963941471430672423450, −10.63449029341170725255055182436, −9.207512407266829742726329163147, −7.84953952415160269306839319128, −7.26383574079510296661067930640, −5.94325921455770750369683453071, −4.60726423679691831073619461850, −3.74600893608931983879658276785, −2.86313216050129286802573859413, −1.90503948069574463657394291788,
1.90503948069574463657394291788, 2.86313216050129286802573859413, 3.74600893608931983879658276785, 4.60726423679691831073619461850, 5.94325921455770750369683453071, 7.26383574079510296661067930640, 7.84953952415160269306839319128, 9.207512407266829742726329163147, 10.63449029341170725255055182436, 11.50886154963941471430672423450
