L(s) = 1 | + (−700. − 404. i)5-s + (−641. − 2.31e3i)7-s + (5.97e3 − 3.45e3i)11-s − 3.31e4·13-s + (6.16e4 − 3.55e4i)17-s + (9.11e4 − 1.57e5i)19-s + (−4.39e5 − 2.53e5i)23-s + (1.31e5 + 2.27e5i)25-s − 1.28e6i·29-s + (−2.20e5 − 3.82e5i)31-s + (−4.86e5 + 1.87e6i)35-s + (1.41e6 − 2.44e6i)37-s − 2.61e6i·41-s + 5.88e6·43-s + (3.57e6 + 2.06e6i)47-s + ⋯ |
L(s) = 1 | + (−1.12 − 0.646i)5-s + (−0.267 − 0.963i)7-s + (0.408 − 0.235i)11-s − 1.16·13-s + (0.737 − 0.426i)17-s + (0.699 − 1.21i)19-s + (−1.56 − 0.906i)23-s + (0.336 + 0.582i)25-s − 1.80i·29-s + (−0.238 − 0.413i)31-s + (−0.323 + 1.25i)35-s + (0.752 − 1.30i)37-s − 0.923i·41-s + 1.72·43-s + (0.732 + 0.422i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.8093091271\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8093091271\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (641. + 2.31e3i)T \) |
good | 5 | \( 1 + (700. + 404. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-5.97e3 + 3.45e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 3.31e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-6.16e4 + 3.55e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-9.11e4 + 1.57e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (4.39e5 + 2.53e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 1.28e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (2.20e5 + 3.82e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.41e6 + 2.44e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 2.61e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.88e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.57e6 - 2.06e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (1.19e7 - 6.88e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (2.24e6 - 1.29e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (6.73e6 - 1.16e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.29e7 - 2.24e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.46e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.86e7 + 3.23e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.45e7 - 2.51e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 4.80e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (1.67e7 + 9.66e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.62e8T + 7.83e15T^{2} \) |
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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
−9.931631448862939569110209091424, −9.111015641639171016482916498658, −7.68652836928699545551875610451, −7.48526330278724793002760474550, −5.97587421718842929741106032937, −4.51542472683588132119631100138, −4.00823196236447473523844977465, −2.57670166643278685685218470269, −0.71388453829560753931356457126, −0.26190455386189546652476259662,
1.58501626828043687216305994470, 2.99286274654673155972515762791, 3.79738521504562238638052588620, 5.19289548159802064483286854424, 6.28789025469843020807703806535, 7.48677811902234246186763148044, 8.064631735245426307170813006123, 9.415867191991012261678949960406, 10.17270232651394149911198526930, 11.43875714524441904957957959108