L(s) = 1 | + (−2.22 + 2.22i)3-s + (−2 − 2i)7-s − 6.89i·9-s + 3.44i·11-s + (−4.44 − 4.44i)13-s + (−0.775 + 0.775i)17-s + 2.55·19-s + 8.89·21-s + (−0.449 + 0.449i)23-s + (8.67 + 8.67i)27-s + 2.89i·29-s − 2.89i·31-s + (−7.67 − 7.67i)33-s + (6 − 6i)37-s + 19.7·39-s + ⋯ |
L(s) = 1 | + (−1.28 + 1.28i)3-s + (−0.755 − 0.755i)7-s − 2.29i·9-s + 1.04i·11-s + (−1.23 − 1.23i)13-s + (−0.188 + 0.188i)17-s + 0.585·19-s + 1.94·21-s + (−0.0937 + 0.0937i)23-s + (1.66 + 1.66i)27-s + 0.538i·29-s − 0.520i·31-s + (−1.33 − 1.33i)33-s + (0.986 − 0.986i)37-s + 3.17·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6798213259\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6798213259\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (2.22 - 2.22i)T - 3iT^{2} \) |
| 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.44iT - 11T^{2} \) |
| 13 | \( 1 + (4.44 + 4.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.775 - 0.775i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.55T + 19T^{2} \) |
| 23 | \( 1 + (0.449 - 0.449i)T - 23iT^{2} \) |
| 29 | \( 1 - 2.89iT - 29T^{2} \) |
| 31 | \( 1 + 2.89iT - 31T^{2} \) |
| 37 | \( 1 + (-6 + 6i)T - 37iT^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + (2.89 - 2.89i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.44 - 6.44i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 - 0.898T + 61T^{2} \) |
| 67 | \( 1 + (4.22 + 4.22i)T + 67iT^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + (-5.67 - 5.67i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.898T + 79T^{2} \) |
| 83 | \( 1 + (4.67 - 4.67i)T - 83iT^{2} \) |
| 89 | \( 1 - 11.8iT - 89T^{2} \) |
| 97 | \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \) |
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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
−9.622214095009739007013205063083, −9.370745157865269852378816294016, −7.71474260984847764271635888698, −7.08786726759755931357954925202, −6.08987986884258970862303418479, −5.37602011770929950985856289247, −4.57369418392983996212544053728, −3.91689602141406857303457998534, −2.80395355190639347462902821278, −0.66394515891896171358083144374,
0.53714113357900642819312602688, 1.95591485598904206856418492912, 2.89901667049167228520355124873, 4.51738884655862349558140498924, 5.48803813828730591182458181258, 6.05397273512797337359265838590, 6.79301250209711840640584163462, 7.35254241838665144671351687620, 8.360252636972400012880584853287, 9.266950343539649070270945877580