L(s) = 1 | + (0.707 + 0.707i)2-s + (1.59 − 0.673i)3-s + 1.00i·4-s + (1.55 − 1.60i)5-s + (1.60 + 0.652i)6-s + (−0.700 + 0.700i)7-s + (−0.707 + 0.707i)8-s + (2.09 − 2.14i)9-s + (2.23 − 0.0306i)10-s − 4.42i·11-s + (0.673 + 1.59i)12-s + (0.282 + 0.282i)13-s − 0.990·14-s + (1.40 − 3.60i)15-s − 1.00·16-s + (1.25 + 1.25i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.921 − 0.388i)3-s + 0.500i·4-s + (0.697 − 0.716i)5-s + (0.655 + 0.266i)6-s + (−0.264 + 0.264i)7-s + (−0.250 + 0.250i)8-s + (0.697 − 0.716i)9-s + (0.707 − 0.00969i)10-s − 1.33i·11-s + (0.194 + 0.460i)12-s + (0.0783 + 0.0783i)13-s − 0.264·14-s + (0.363 − 0.931i)15-s − 0.250·16-s + (0.305 + 0.305i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.09363 - 0.217995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.09363 - 0.217995i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.59 + 0.673i)T \) |
| 5 | \( 1 + (-1.55 + 1.60i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (0.700 - 0.700i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.42iT - 11T^{2} \) |
| 13 | \( 1 + (-0.282 - 0.282i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.25 - 1.25i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.10iT - 19T^{2} \) |
| 23 | \( 1 + (-3.17 + 3.17i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.862T + 29T^{2} \) |
| 37 | \( 1 + (1.12 - 1.12i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.40iT - 41T^{2} \) |
| 43 | \( 1 + (5.61 + 5.61i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.96 - 5.96i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.34 - 7.34i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.852T + 59T^{2} \) |
| 61 | \( 1 + 3.32T + 61T^{2} \) |
| 67 | \( 1 + (-7.00 + 7.00i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.54iT - 71T^{2} \) |
| 73 | \( 1 + (8.92 + 8.92i)T + 73iT^{2} \) |
| 79 | \( 1 - 2.12iT - 79T^{2} \) |
| 83 | \( 1 + (3.12 - 3.12i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 + (11.2 - 11.2i)T - 97iT^{2} \) |
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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
−9.758852558490959232177921686468, −8.959091289868166379427547315397, −8.381048200116650311156597764560, −7.67901003617816217102410465093, −6.33806805029762310139727909859, −5.97980903776200434208125712575, −4.78469174755245294382132974240, −3.62753455090804047031174161185, −2.73325794132892970803593814024, −1.32054067528193721629189323031,
1.78389719445113728669972796827, 2.69715749801656804077770091531, 3.55255106614923740631042233192, 4.63329700930252677786472525812, 5.48069785474926611791670416450, 6.96522873219627245246114214425, 7.21550039967988563165891595848, 8.679532304798380801575636488777, 9.625636549570520208614194255152, 9.930689515598453471223393973980