L(s) = 1 | + (1.22 + 1.22i)2-s + (−1.22 + 1.22i)3-s − 1.00i·4-s − 2.99·6-s + (−4.89 − 4.89i)7-s + (6.12 − 6.12i)8-s + 6i·9-s − 3·11-s + (1.22 + 1.22i)12-s + (−7.34 + 7.34i)13-s − 11.9i·14-s + 10.9·16-s + (13.4 + 13.4i)17-s + (−7.34 + 7.34i)18-s + 5i·19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.612i)2-s + (−0.408 + 0.408i)3-s − 0.250i·4-s − 0.499·6-s + (−0.699 − 0.699i)7-s + (0.765 − 0.765i)8-s + 0.666i·9-s − 0.272·11-s + (0.102 + 0.102i)12-s + (−0.565 + 0.565i)13-s − 0.857i·14-s + 0.687·16-s + (0.792 + 0.792i)17-s + (−0.408 + 0.408i)18-s + 0.263i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.00672 + 0.341600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00672 + 0.341600i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-1.22 - 1.22i)T + 4iT^{2} \) |
| 3 | \( 1 + (1.22 - 1.22i)T - 9iT^{2} \) |
| 7 | \( 1 + (4.89 + 4.89i)T + 49iT^{2} \) |
| 11 | \( 1 + 3T + 121T^{2} \) |
| 13 | \( 1 + (7.34 - 7.34i)T - 169iT^{2} \) |
| 17 | \( 1 + (-13.4 - 13.4i)T + 289iT^{2} \) |
| 19 | \( 1 - 5iT - 361T^{2} \) |
| 23 | \( 1 + (-17.1 + 17.1i)T - 529iT^{2} \) |
| 29 | \( 1 + 30iT - 841T^{2} \) |
| 31 | \( 1 + 38T + 961T^{2} \) |
| 37 | \( 1 + (-19.5 - 19.5i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 57T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-4.89 + 4.89i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-7.34 - 7.34i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (31.8 - 31.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 90iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28T + 3.72e3T^{2} \) |
| 67 | \( 1 + (47.7 + 47.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 42T + 5.04e3T^{2} \) |
| 73 | \( 1 + (13.4 - 13.4i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 80iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (111. - 111. i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 15iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (53.8 + 53.8i)T + 9.40e3iT^{2} \) |
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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
−16.85019171884077818291484826468, −16.37859451314354405342318734091, −15.05651061438866677572858832335, −13.91243975330688633400572398253, −12.77679944275163605300715426036, −10.80202217950719097990488636664, −9.843700278017067399731181305591, −7.43070867433979855000875174944, −5.90391852040235819298130691681, −4.38168152664509159724498301141,
3.11856284495437891957650319790, 5.45386990868911896353903542376, 7.36806039937676854566796043849, 9.354066587400336108683727151781, 11.20975637095062978438764001943, 12.39871966517840062865009927183, 12.92108624713706615057147601345, 14.56750584768152811512563579792, 16.06656954705346165557575563172, 17.37879555976911716170250167333