L(s) = 1 | + (−2.66 − 2.66i)2-s + (−2.12 − 2.12i)3-s + 10.1i·4-s + (−4.54 − 2.09i)5-s + 11.3i·6-s + (2.95 + 2.95i)7-s + (16.4 − 16.4i)8-s + 0.0323i·9-s + (6.51 + 17.6i)10-s + 8.43i·11-s + (21.6 − 21.6i)12-s + (−9.88 + 8.44i)13-s − 15.7i·14-s + (5.19 + 14.0i)15-s − 47.1·16-s + (−9.34 + 9.34i)17-s + ⋯ |
L(s) = 1 | + (−1.33 − 1.33i)2-s + (−0.708 − 0.708i)3-s + 2.54i·4-s + (−0.908 − 0.418i)5-s + 1.88i·6-s + (0.421 + 0.421i)7-s + (2.06 − 2.06i)8-s + 0.00359i·9-s + (0.651 + 1.76i)10-s + 0.767i·11-s + (1.80 − 1.80i)12-s + (−0.760 + 0.649i)13-s − 1.12i·14-s + (0.346 + 0.939i)15-s − 2.94·16-s + (−0.549 + 0.549i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00134578 + 0.00101429i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00134578 + 0.00101429i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (4.54 + 2.09i)T \) |
| 13 | \( 1 + (9.88 - 8.44i)T \) |
good | 2 | \( 1 + (2.66 + 2.66i)T + 4iT^{2} \) |
| 3 | \( 1 + (2.12 + 2.12i)T + 9iT^{2} \) |
| 7 | \( 1 + (-2.95 - 2.95i)T + 49iT^{2} \) |
| 11 | \( 1 - 8.43iT - 121T^{2} \) |
| 17 | \( 1 + (9.34 - 9.34i)T - 289iT^{2} \) |
| 19 | \( 1 + 12.7T + 361T^{2} \) |
| 23 | \( 1 + (28.0 + 28.0i)T + 529iT^{2} \) |
| 29 | \( 1 - 32.7iT - 841T^{2} \) |
| 31 | \( 1 + 36.2iT - 961T^{2} \) |
| 37 | \( 1 + (-17.1 - 17.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 26.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (34.4 + 34.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (9.09 + 9.09i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-31.0 - 31.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 68.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 30.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (5.29 + 5.29i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 2.71iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (29.3 - 29.3i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 12.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (3.27 - 3.27i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 68.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-51.9 - 51.9i)T + 9.40e3iT^{2} \) |
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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
−12.84593382398430740638230558834, −12.16375362393841906088394036946, −11.67619246132775874947098653709, −10.52669683021708532236779298968, −9.125438499742066640183789922326, −8.114257015208009374088432381950, −6.94740817316902363701417818197, −4.26691882527784982802209492212, −1.94798964310347747053721463676, −0.00248442147202340548504194465,
4.69179238155416898585358388298, 6.04560361595719222066099286212, 7.47617711254723579704341165162, 8.236230112075606169587075681487, 9.793990184345056650434235593515, 10.71230555296395248135331407702, 11.49172666221858221643185879379, 13.91914097699901284935100239054, 15.04689026251565901054198601490, 15.79689079941501870290522811095