| L(s) = 1 | − 5.93·2-s − 7.05·3-s + 3.20·4-s − 25·5-s + 41.8·6-s − 185.·7-s + 170.·8-s − 193.·9-s + 148.·10-s − 353.·11-s − 22.5·12-s + 169·13-s + 1.10e3·14-s + 176.·15-s − 1.11e3·16-s + 634.·17-s + 1.14e3·18-s + 1.11e3·19-s − 80.0·20-s + 1.31e3·21-s + 2.09e3·22-s + 3.50e3·23-s − 1.20e3·24-s + 625·25-s − 1.00e3·26-s + 3.07e3·27-s − 594.·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s − 0.452·3-s + 0.100·4-s − 0.447·5-s + 0.474·6-s − 1.43·7-s + 0.943·8-s − 0.795·9-s + 0.469·10-s − 0.881·11-s − 0.0452·12-s + 0.277·13-s + 1.50·14-s + 0.202·15-s − 1.09·16-s + 0.532·17-s + 0.834·18-s + 0.710·19-s − 0.0447·20-s + 0.648·21-s + 0.924·22-s + 1.38·23-s − 0.427·24-s + 0.200·25-s − 0.290·26-s + 0.812·27-s − 0.143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3473950352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3473950352\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 5.93T + 32T^{2} \) |
| 3 | \( 1 + 7.05T + 243T^{2} \) |
| 7 | \( 1 + 185.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 353.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 634.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.50e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 283.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.78e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.66e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.80e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.77e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.65e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.01e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.42e5T + 8.58e9T^{2} \) |
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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
−13.70331332182632108028960899296, −12.73628393180662539520553388342, −11.35790355123319274814812472854, −10.29808429557212255948187219141, −9.302758587356092743256635625421, −8.152734433888632152134105539261, −6.86566879628590454806447003936, −5.28768182067449187744975508917, −3.20507252859091977178613137695, −0.54027887011482223003609123981,
0.54027887011482223003609123981, 3.20507252859091977178613137695, 5.28768182067449187744975508917, 6.86566879628590454806447003936, 8.152734433888632152134105539261, 9.302758587356092743256635625421, 10.29808429557212255948187219141, 11.35790355123319274814812472854, 12.73628393180662539520553388342, 13.70331332182632108028960899296
