| L(s) = 1 | + 4.79·2-s − 2.86·3-s + 15.0·4-s + 0.463·5-s − 13.7·6-s + 1.01·7-s + 33.6·8-s − 18.7·9-s + 2.22·10-s − 21.1·11-s − 42.9·12-s − 44.0·13-s + 4.86·14-s − 1.32·15-s + 41.1·16-s + 108.·17-s − 90.1·18-s + 144.·19-s + 6.95·20-s − 2.90·21-s − 101.·22-s − 56.8·23-s − 96.2·24-s − 124.·25-s − 211.·26-s + 131.·27-s + 15.2·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s − 0.551·3-s + 1.87·4-s + 0.0414·5-s − 0.934·6-s + 0.0547·7-s + 1.48·8-s − 0.696·9-s + 0.0702·10-s − 0.578·11-s − 1.03·12-s − 0.939·13-s + 0.0929·14-s − 0.0228·15-s + 0.643·16-s + 1.54·17-s − 1.18·18-s + 1.73·19-s + 0.0777·20-s − 0.0301·21-s − 0.981·22-s − 0.515·23-s − 0.818·24-s − 0.998·25-s − 1.59·26-s + 0.934·27-s + 0.102·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.408250139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408250139\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 37T \) |
| good | 2 | \( 1 - 4.79T + 8T^{2} \) |
| 3 | \( 1 + 2.86T + 27T^{2} \) |
| 5 | \( 1 - 0.463T + 125T^{2} \) |
| 7 | \( 1 - 1.01T + 343T^{2} \) |
| 11 | \( 1 + 21.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 64.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 206.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 166.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 72.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 532.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 64.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 810.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 639.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 623.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 831.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 94.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 43.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 252.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 606.T + 9.12e5T^{2} \) |
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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
−15.65033384733423409110961268824, −14.41370815294741794067647171857, −13.67451971357289179496513102074, −12.15634064571078012200937994619, −11.72698656845913464068635593052, −10.08976662984497465131658886225, −7.60721700948092894492904470373, −5.89894839889632647515446775532, −5.01821854609445596811497325013, −3.04883537401242207007917974426,
3.04883537401242207007917974426, 5.01821854609445596811497325013, 5.89894839889632647515446775532, 7.60721700948092894492904470373, 10.08976662984497465131658886225, 11.72698656845913464068635593052, 12.15634064571078012200937994619, 13.67451971357289179496513102074, 14.41370815294741794067647171857, 15.65033384733423409110961268824
