| L(s) = 1 | + (−2.23 + 2.23i)3-s + (2.23 + 2.23i)7-s − 7.00i·9-s − 10.0·21-s + (−6.70 + 6.70i)23-s + (8.94 + 8.94i)27-s + 6i·29-s − 12·41-s + (2.23 − 2.23i)43-s + (6.70 + 6.70i)47-s + 3.00i·49-s − 8·61-s + (15.6 − 15.6i)63-s + (−11.1 − 11.1i)67-s − 30.0i·69-s + ⋯ |
| L(s) = 1 | + (−1.29 + 1.29i)3-s + (0.845 + 0.845i)7-s − 2.33i·9-s − 2.18·21-s + (−1.39 + 1.39i)23-s + (1.72 + 1.72i)27-s + 1.11i·29-s − 1.87·41-s + (0.340 − 0.340i)43-s + (0.978 + 0.978i)47-s + 0.428i·49-s − 1.02·61-s + (1.97 − 1.97i)63-s + (−1.36 − 1.36i)67-s − 3.61i·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5049101324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5049101324\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 - 2.23i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.23 - 2.23i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (6.70 - 6.70i)T - 23iT^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + (-2.23 + 2.23i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.70 - 6.70i)T + 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (11.1 + 11.1i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (6.70 - 6.70i)T - 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−9.950511357210543986743211941790, −9.255308550105184062411518224100, −8.516789131338470936017772538959, −7.44514014880009423663181672796, −6.27859281039527853247866922275, −5.59621863950198405819627474859, −5.06137598321172074073776051210, −4.23697490115455824521567072286, −3.28014324249818554111203379071, −1.64089648115142994020607020848,
0.24282192574704404061241435164, 1.37070090535459281878396255306, 2.32498278482084903333985919837, 4.14133005602846080060895961893, 4.89661310534462206726466716050, 5.85962752097497149678492837505, 6.49170703068493345440291620803, 7.31280967934798289946520326900, 7.87710474780366532107216500855, 8.605508034733819342430342404658
