L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (3.44 + 3.44i)7-s + (0.707 + 0.707i)8-s + 4.56i·11-s + (−1.77 + 1.77i)13-s − 4.87·14-s − 1.00·16-s + (−2.75 + 2.75i)17-s + 0.449i·19-s + (−3.22 − 3.22i)22-s + (−5.90 − 5.90i)23-s − 2.51i·26-s + (3.44 − 3.44i)28-s − 0.317·29-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (1.30 + 1.30i)7-s + (0.250 + 0.250i)8-s + 1.37i·11-s + (−0.492 + 0.492i)13-s − 1.30·14-s − 0.250·16-s + (−0.668 + 0.668i)17-s + 0.103i·19-s + (−0.687 − 0.687i)22-s + (−1.23 − 1.23i)23-s − 0.492i·26-s + (0.651 − 0.651i)28-s − 0.0590·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.098768823\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098768823\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-3.44 - 3.44i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.56iT - 11T^{2} \) |
| 13 | \( 1 + (1.77 - 1.77i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.75 - 2.75i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.449iT - 19T^{2} \) |
| 23 | \( 1 + (5.90 + 5.90i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.317T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 2.04iT - 41T^{2} \) |
| 43 | \( 1 + (-4.77 + 4.77i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.07 - 3.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.65 + 5.65i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + (2 + 2i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.142iT - 71T^{2} \) |
| 73 | \( 1 + (0.449 - 0.449i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.550iT - 79T^{2} \) |
| 83 | \( 1 + (-6.75 - 6.75i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 + (3.55 + 3.55i)T + 97iT^{2} \) |
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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
−9.764029333159197640817864771854, −9.007934525684119876813548513569, −8.288244838550912356860919420608, −7.71814104019975980726736407384, −6.68143197683421589708969984705, −5.91132403497236849470929120610, −4.84692428656863703531079056157, −4.39092560010319759410756170889, −2.32771608149826629124919065923, −1.81825879726442193430924481379,
0.52779223233352063121257737890, 1.65734278829974927085284046463, 3.00683140865823739599836698352, 4.02767483035348595927705602457, 4.85355085793871976857207548287, 5.93624741801968655492598935415, 7.16446564186203239116326663578, 7.82644693697916288880234945020, 8.345170897641611833252910908038, 9.341388036595284602634385533656