| L(s) = 1 | + (−0.256 + 0.306i)2-s + (−0.0473 + 0.0397i)3-s + (0.319 + 1.81i)4-s + (0.853 − 2.34i)5-s − 0.0247i·6-s + (−4.17 − 1.52i)7-s + (−1.32 − 0.767i)8-s + (−0.520 + 2.95i)9-s + (0.498 + 0.863i)10-s + (1.37 − 2.37i)11-s + (−0.0871 − 0.0731i)12-s + (0.327 − 0.0577i)13-s + (1.53 − 0.888i)14-s + (0.0527 + 0.145i)15-s + (−2.88 + 1.04i)16-s + (4.31 + 0.760i)17-s + ⋯ |
| L(s) = 1 | + (−0.181 + 0.216i)2-s + (−0.0273 + 0.0229i)3-s + (0.159 + 0.906i)4-s + (0.381 − 1.04i)5-s − 0.0100i·6-s + (−1.57 − 0.574i)7-s + (−0.469 − 0.271i)8-s + (−0.173 + 0.983i)9-s + (0.157 + 0.273i)10-s + (0.413 − 0.715i)11-s + (−0.0251 − 0.0211i)12-s + (0.0908 − 0.0160i)13-s + (0.411 − 0.237i)14-s + (0.0136 + 0.0374i)15-s + (−0.720 + 0.262i)16-s + (1.04 + 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.670721 + 0.112897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.670721 + 0.112897i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-5.31 - 2.96i)T \) |
| good | 2 | \( 1 + (0.256 - 0.306i)T + (-0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (0.0473 - 0.0397i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.853 + 2.34i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (4.17 + 1.52i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.37 + 2.37i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.327 + 0.0577i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.31 - 0.760i)T + (15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-2.41 - 2.87i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (3.45 - 1.99i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.90 + 2.25i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.06iT - 31T^{2} \) |
| 41 | \( 1 + (1.71 + 9.71i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 5.17iT - 43T^{2} \) |
| 47 | \( 1 + (2.89 + 5.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.30 - 1.56i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-4.68 - 12.8i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.55 + 0.274i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.41 + 1.60i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.88 + 2.42i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 8.59T + 73T^{2} \) |
| 79 | \( 1 + (-2.58 + 7.09i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.340 - 1.92i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (4.13 + 11.3i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.861 - 0.497i)T + (48.5 - 84.0i)T^{2} \) |
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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
−16.57796354917088395117350447863, −16.01673128386642303323854625500, −13.75145725567467945893988972591, −13.02871129541994977255401859292, −11.94943776280352833023886044230, −10.09780049546779024148332772660, −8.865151217600835051101812100734, −7.55018295027993519104210222436, −5.85678719983371844541762279429, −3.58718838543741772423102545855,
2.97825955296621525725423276437, 6.00104516189345373166188731961, 6.75311494171520517381752169527, 9.524399660054173032294095224817, 9.829578522562252980620971878866, 11.43150419064730034310347257856, 12.69204658908926598288629042137, 14.34718775219012087995025866959, 15.04798336809554861124396464573, 16.20230056830266442096849751675
