L(s) = 1 | + (0.707 − 0.707i)2-s + (1.53 + 0.796i)3-s − 1.00i·4-s + (−2.02 + 2.02i)5-s + (1.65 − 0.524i)6-s + (−2.53 + 2.53i)7-s + (−0.707 − 0.707i)8-s + (1.73 + 2.44i)9-s + 2.85i·10-s + (−2.97 − 2.97i)11-s + (0.796 − 1.53i)12-s + 3.58i·14-s + (−4.71 + 1.49i)15-s − 1.00·16-s − 3.45·17-s + (2.95 + 0.507i)18-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.888 + 0.459i)3-s − 0.500i·4-s + (−0.903 + 0.903i)5-s + (0.673 − 0.214i)6-s + (−0.959 + 0.959i)7-s + (−0.250 − 0.250i)8-s + (0.577 + 0.816i)9-s + 0.903i·10-s + (−0.895 − 0.895i)11-s + (0.229 − 0.444i)12-s + 0.959i·14-s + (−1.21 + 0.387i)15-s − 0.250·16-s − 0.837·17-s + (0.696 + 0.119i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.180994586\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.180994586\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.53 - 0.796i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (2.02 - 2.02i)T - 5iT^{2} \) |
| 7 | \( 1 + (2.53 - 2.53i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.97 + 2.97i)T + 11iT^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 19 | \( 1 + (-1.58 - 1.58i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.03T + 23T^{2} \) |
| 29 | \( 1 - 2.01iT - 29T^{2} \) |
| 31 | \( 1 + (-1.21 - 1.21i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.42 - 3.42i)T - 37iT^{2} \) |
| 41 | \( 1 + (5.61 - 5.61i)T - 41iT^{2} \) |
| 43 | \( 1 + 1.95iT - 43T^{2} \) |
| 47 | \( 1 + (-0.957 - 0.957i)T + 47iT^{2} \) |
| 53 | \( 1 - 7.22iT - 53T^{2} \) |
| 59 | \( 1 + (-7.27 - 7.27i)T + 59iT^{2} \) |
| 61 | \( 1 - 0.274T + 61T^{2} \) |
| 67 | \( 1 + (3.00 + 3.00i)T + 67iT^{2} \) |
| 71 | \( 1 + (-7.80 + 7.80i)T - 71iT^{2} \) |
| 73 | \( 1 + (-10.0 + 10.0i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.58T + 79T^{2} \) |
| 83 | \( 1 + (-2.58 + 2.58i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.96 - 6.96i)T + 89iT^{2} \) |
| 97 | \( 1 + (5.67 + 5.67i)T + 97iT^{2} \) |
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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
−10.37636023370316628120636968244, −9.496543046144764438646439742131, −8.645940734829873054995786974704, −7.897502094221173701674831156030, −6.84847629244953535724763303984, −5.89930986450713182573863615675, −4.81102322815284291849896667847, −3.56314158094782607573997588964, −3.14678739581039651715913882433, −2.32157847997563370574835105988,
0.37928590595796574531052600358, 2.29894963292534018009085784621, 3.61183861094997483353919175294, 4.18247153205392232829605506922, 5.14214085133448634333350362829, 6.62399581003249874369679687755, 7.17453221864954368173607888554, 7.913997133683928330279302053181, 8.556661131677586854199823293595, 9.529310421515146656043970574185