L(s) = 1 | + (−0.446 + 0.119i)2-s + (−1.54 + 0.893i)4-s + (2.09 + 0.562i)5-s + (−1.26 + 2.32i)7-s + (1.23 − 1.23i)8-s − 1.00·10-s + (4.08 − 4.08i)11-s + (2.90 − 2.13i)13-s + (0.288 − 1.18i)14-s + (1.38 − 2.39i)16-s + (−0.0614 − 0.106i)17-s + (−2.29 + 2.29i)19-s + (−3.74 + 1.00i)20-s + (−1.33 + 2.31i)22-s + (2.23 + 1.28i)23-s + ⋯ |
L(s) = 1 | + (−0.315 + 0.0845i)2-s + (−0.773 + 0.446i)4-s + (0.938 + 0.251i)5-s + (−0.479 + 0.877i)7-s + (0.437 − 0.437i)8-s − 0.317·10-s + (1.23 − 1.23i)11-s + (0.805 − 0.592i)13-s + (0.0772 − 0.317i)14-s + (0.345 − 0.598i)16-s + (−0.0149 − 0.0258i)17-s + (−0.526 + 0.526i)19-s + (−0.838 + 0.224i)20-s + (−0.284 + 0.492i)22-s + (0.465 + 0.268i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23157 + 0.537311i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23157 + 0.537311i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.26 - 2.32i)T \) |
| 13 | \( 1 + (-2.90 + 2.13i)T \) |
good | 2 | \( 1 + (0.446 - 0.119i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-2.09 - 0.562i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-4.08 + 4.08i)T - 11iT^{2} \) |
| 17 | \( 1 + (0.0614 + 0.106i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.29 - 2.29i)T - 19iT^{2} \) |
| 23 | \( 1 + (-2.23 - 1.28i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.15 - 7.20i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.940 - 3.50i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.75 - 6.55i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.63 - 0.707i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.72 - 2.14i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.591 - 2.20i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.49 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.94 + 7.26i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 6.15iT - 61T^{2} \) |
| 67 | \( 1 + (7.99 + 7.99i)T + 67iT^{2} \) |
| 71 | \( 1 + (7.44 - 1.99i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-12.0 + 3.23i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.76 - 4.76i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.966 - 0.259i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.38 - 8.88i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.13728817937871497422902897266, −9.332665463472233345016930350318, −8.735446347494422094466700006886, −8.181666062035271283471847849363, −6.65061303239371874574526382117, −6.09105549483413683155403444510, −5.17607433450972000614691395368, −3.73698855482656832006393917936, −2.96326497176198659607020950400, −1.21921634182322962085855639096,
0.966953999618576911100682137309, 2.08047497014489970936030754337, 4.07598948379569185011721984221, 4.46334794019696155673134567632, 5.86540500362518263648468610794, 6.54494939253718204278593352939, 7.53640225751019509861308961352, 8.888028145478267482184445408398, 9.288080217404665039305999325862, 9.975167062923858041657469180771