| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 5-s − 7-s − 2.23·8-s + 0.618·10-s − 0.236·11-s + 3.61·13-s − 0.618·14-s + 1.85·16-s − 0.618·17-s + 4.09·19-s − 1.61·20-s − 0.145·22-s + 7.61·23-s + 25-s + 2.23·26-s + 1.61·28-s + 10.5·29-s − 6.70·31-s + 5.61·32-s − 0.381·34-s − 35-s − 6.70·37-s + 2.52·38-s − 2.23·40-s − 3.09·41-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.447·5-s − 0.377·7-s − 0.790·8-s + 0.195·10-s − 0.0711·11-s + 1.00·13-s − 0.165·14-s + 0.463·16-s − 0.149·17-s + 0.938·19-s − 0.361·20-s − 0.0311·22-s + 1.58·23-s + 0.200·25-s + 0.438·26-s + 0.305·28-s + 1.96·29-s − 1.20·31-s + 0.993·32-s − 0.0655·34-s − 0.169·35-s − 1.10·37-s + 0.410·38-s − 0.353·40-s − 0.482·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.717700806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717700806\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 - 11T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 7.14T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 + 2.85T + 71T^{2} \) |
| 73 | \( 1 + 3.94T + 73T^{2} \) |
| 79 | \( 1 - 3.38T + 79T^{2} \) |
| 83 | \( 1 + 2.70T + 83T^{2} \) |
| 89 | \( 1 + 7.47T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−10.04042920231119381932690737441, −8.959351555927812288297300214900, −8.803396689326762435839335571011, −7.44588427785290898991507802330, −6.44694501343437403443347738583, −5.58652582296366593975178811046, −4.84291327957264541803336986666, −3.71676745370705737880485753203, −2.86809851654240131352188138538, −1.03363887182943800339364220791,
1.03363887182943800339364220791, 2.86809851654240131352188138538, 3.71676745370705737880485753203, 4.84291327957264541803336986666, 5.58652582296366593975178811046, 6.44694501343437403443347738583, 7.44588427785290898991507802330, 8.803396689326762435839335571011, 8.959351555927812288297300214900, 10.04042920231119381932690737441
