L(s) = 1 | − 2-s − 0.987i·3-s + 4-s + (−1.85 − 1.25i)5-s + 0.987i·6-s + 4.78i·7-s − 8-s + 2.02·9-s + (1.85 + 1.25i)10-s − 5.98·11-s − 0.987i·12-s − 3.49·13-s − 4.78i·14-s + (−1.23 + 1.83i)15-s + 16-s + 4.96·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.570i·3-s + 0.5·4-s + (−0.829 − 0.559i)5-s + 0.403i·6-s + 1.81i·7-s − 0.353·8-s + 0.674·9-s + (0.586 + 0.395i)10-s − 1.80·11-s − 0.285i·12-s − 0.969·13-s − 1.28i·14-s + (−0.318 + 0.472i)15-s + 0.250·16-s + 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.318311 + 0.357700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.318311 + 0.357700i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (1.85 + 1.25i)T \) |
| 37 | \( 1 + (3.96 + 4.61i)T \) |
good | 3 | \( 1 + 0.987iT - 3T^{2} \) |
| 7 | \( 1 - 4.78iT - 7T^{2} \) |
| 11 | \( 1 + 5.98T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 - 7.33iT - 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 - 7.85iT - 29T^{2} \) |
| 31 | \( 1 - 3.24iT - 31T^{2} \) |
| 41 | \( 1 + 0.530T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 - 4.30iT - 47T^{2} \) |
| 53 | \( 1 - 3.66iT - 53T^{2} \) |
| 59 | \( 1 + 2.15iT - 59T^{2} \) |
| 61 | \( 1 + 3.06iT - 61T^{2} \) |
| 67 | \( 1 - 3.79iT - 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 + 9.05iT - 73T^{2} \) |
| 79 | \( 1 + 5.56iT - 79T^{2} \) |
| 83 | \( 1 - 3.77iT - 83T^{2} \) |
| 89 | \( 1 + 8.45iT - 89T^{2} \) |
| 97 | \( 1 - 3.64T + 97T^{2} \) |
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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
−11.99996588639874897453087380335, −10.58605587936135590374427997377, −9.763529889530567511027801928270, −8.669697033091554326533298478998, −7.899331499897352351985618520508, −7.40711452706421579955494486887, −5.79194675102613954828463105359, −5.05442346435072575419464349704, −3.09165077297633279601414564889, −1.80734505940661399190237106758,
0.39426706405684723631154924903, 2.80636339663185185620768448702, 4.04688652980808904983934830242, 5.00788148222607807125689255783, 6.91033838978373739267780471011, 7.51124458316273504593600372692, 8.019096571760212417117743515480, 9.764287874579912045986869459551, 10.24419134363808840282938587712, 10.75848415588050398522903308679