L(s) = 1 | + (1.08 + 2.61i)2-s + (−5.14 − 7.69i)3-s + (−5.65 + 5.65i)4-s + (14.5 − 21.7i)6-s + (−20.9 − 8.65i)8-s + (−22.4 + 54.2i)9-s + (57.2 + 38.2i)11-s + (72.6 + 14.4i)12-s − 64i·16-s + (−32.4 + 62.1i)17-s − 165.·18-s + (63.1 + 152. i)19-s + (−37.9 + 190. i)22-s + (40.8 + 205. i)24-s + (−115. − 47.8i)25-s + ⋯ |
L(s) = 1 | + (0.382 + 0.923i)2-s + (−0.989 − 1.48i)3-s + (−0.707 + 0.707i)4-s + (0.989 − 1.48i)6-s + (−0.923 − 0.382i)8-s + (−0.831 + 2.00i)9-s + (1.56 + 1.04i)11-s + (1.74 + 0.347i)12-s − i·16-s + (−0.462 + 0.886i)17-s − 2.17·18-s + (0.762 + 1.84i)19-s + (−0.367 + 1.84i)22-s + (0.347 + 1.74i)24-s + (−0.923 − 0.382i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.683136 + 0.786055i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.683136 + 0.786055i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.08 - 2.61i)T \) |
| 17 | \( 1 + (32.4 - 62.1i)T \) |
good | 3 | \( 1 + (5.14 + 7.69i)T + (-10.3 + 24.9i)T^{2} \) |
| 5 | \( 1 + (115. + 47.8i)T^{2} \) |
| 7 | \( 1 + (316. - 131. i)T^{2} \) |
| 11 | \( 1 + (-57.2 - 38.2i)T + (509. + 1.22e3i)T^{2} \) |
| 13 | \( 1 - 2.19e3iT^{2} \) |
| 19 | \( 1 + (-63.1 - 152. i)T + (-4.85e3 + 4.85e3i)T^{2} \) |
| 23 | \( 1 + (4.65e3 + 1.12e4i)T^{2} \) |
| 29 | \( 1 + (-2.25e4 - 9.33e3i)T^{2} \) |
| 31 | \( 1 + (-1.14e4 + 2.75e4i)T^{2} \) |
| 37 | \( 1 + (1.93e4 - 4.67e4i)T^{2} \) |
| 41 | \( 1 + (-78.5 - 394. i)T + (-6.36e4 + 2.63e4i)T^{2} \) |
| 43 | \( 1 + (-128. + 310. i)T + (-5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 - 1.03e5iT^{2} \) |
| 53 | \( 1 + (1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (-21.4 - 8.89i)T + (1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (-2.09e5 + 8.68e4i)T^{2} \) |
| 67 | \( 1 - 354. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + (1.36e5 - 3.30e5i)T^{2} \) |
| 73 | \( 1 + (236. - 1.18e3i)T + (-3.59e5 - 1.48e5i)T^{2} \) |
| 79 | \( 1 + (-1.88e5 - 4.55e5i)T^{2} \) |
| 83 | \( 1 + (911. - 377. i)T + (4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-1.14e3 + 1.14e3i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + (414. + 82.4i)T + (8.43e5 + 3.49e5i)T^{2} \) |
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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
−12.88705303818538245433572108734, −12.22901810524094427609697257819, −11.58914352449361199502852137345, −9.809943199311410189453528500367, −8.270850619497227818501961087360, −7.33488983558121008560257808952, −6.44802996072948594853744307506, −5.73169398794425497615456215253, −4.14917825917106459293187340301, −1.54479769273759013644577738233,
0.56766301551703176094450394565, 3.28765422127457147600573518406, 4.34653870923956831555817930934, 5.34386484977851186144251965184, 6.44891889179004582804823899570, 9.184790100860610991665195406794, 9.333891749564327573740927921505, 10.73292603979127529507261549529, 11.45962186901894779412168366591, 11.83210802134313600809035832056