| L(s) = 1 | + 4.92·2-s + 1.54·3-s + 16.2·4-s − 14.7·5-s + 7.62·6-s + 40.8·8-s − 24.6·9-s − 72.5·10-s − 53.0·11-s + 25.1·12-s + 26.8·13-s − 22.7·15-s + 70.9·16-s + 17.1·17-s − 121.·18-s + 119.·19-s − 239.·20-s − 261.·22-s + 180.·23-s + 63.1·24-s + 91.9·25-s + 132.·26-s − 79.8·27-s − 49.8·29-s − 112.·30-s + 11.0·31-s + 22.9·32-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.297·3-s + 2.03·4-s − 1.31·5-s + 0.518·6-s + 1.80·8-s − 0.911·9-s − 2.29·10-s − 1.45·11-s + 0.606·12-s + 0.572·13-s − 0.392·15-s + 1.10·16-s + 0.244·17-s − 1.58·18-s + 1.44·19-s − 2.68·20-s − 2.53·22-s + 1.64·23-s + 0.537·24-s + 0.735·25-s + 0.997·26-s − 0.569·27-s − 0.318·29-s − 0.683·30-s + 0.0637·31-s + 0.126·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.099085935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.099085935\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 4.92T + 8T^{2} \) |
| 3 | \( 1 - 1.54T + 27T^{2} \) |
| 5 | \( 1 + 14.7T + 125T^{2} \) |
| 11 | \( 1 + 53.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 49.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 11.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.20T + 5.06e4T^{2} \) |
| 41 | \( 1 - 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 49.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 486.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 198.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 664.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 498.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 680.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 596.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 8.05T + 4.93e5T^{2} \) |
| 83 | \( 1 - 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 863.T + 9.12e5T^{2} \) |
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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
−8.346746353001418373854377115026, −7.51683636394367399830117522532, −7.19622952792118282516206075494, −5.85846706723367750161619331411, −5.39248684837925913902169436452, −4.61249567792863311240574784768, −3.67372997775796230104088591979, −3.10534059941311175664171146420, −2.49388548311218357867310333216, −0.73137653125172707512644064863,
0.73137653125172707512644064863, 2.49388548311218357867310333216, 3.10534059941311175664171146420, 3.67372997775796230104088591979, 4.61249567792863311240574784768, 5.39248684837925913902169436452, 5.85846706723367750161619331411, 7.19622952792118282516206075494, 7.51683636394367399830117522532, 8.346746353001418373854377115026
