L(s) = 1 | + i·2-s + 1.78i·3-s − 4-s + (−0.264 + 2.22i)5-s − 1.78·6-s + 1.70i·7-s − i·8-s − 0.174·9-s + (−2.22 − 0.264i)10-s − 0.955·11-s − 1.78i·12-s − 3.25i·13-s − 1.70·14-s + (−3.95 − 0.471i)15-s + 16-s + 0.825i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.02i·3-s − 0.5·4-s + (−0.118 + 0.992i)5-s − 0.727·6-s + 0.643i·7-s − 0.353i·8-s − 0.0580·9-s + (−0.702 − 0.0836i)10-s − 0.288·11-s − 0.514i·12-s − 0.902i·13-s − 0.455·14-s + (−1.02 − 0.121i)15-s + 0.250·16-s + 0.200i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0668458 + 1.12653i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0668458 + 1.12653i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (0.264 - 2.22i)T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 - 1.78iT - 3T^{2} \) |
| 7 | \( 1 - 1.70iT - 7T^{2} \) |
| 11 | \( 1 + 0.955T + 11T^{2} \) |
| 13 | \( 1 + 3.25iT - 13T^{2} \) |
| 17 | \( 1 - 0.825iT - 17T^{2} \) |
| 19 | \( 1 + 0.877T + 19T^{2} \) |
| 23 | \( 1 - 0.580iT - 23T^{2} \) |
| 29 | \( 1 - 3.83T + 29T^{2} \) |
| 37 | \( 1 - 2.21iT - 37T^{2} \) |
| 41 | \( 1 - 3.33T + 41T^{2} \) |
| 43 | \( 1 + 7.87iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 2.19iT - 53T^{2} \) |
| 59 | \( 1 - 0.464T + 59T^{2} \) |
| 61 | \( 1 - 2.77T + 61T^{2} \) |
| 67 | \( 1 - 14.5iT - 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 + 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 1.56T + 79T^{2} \) |
| 83 | \( 1 - 9.44iT - 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−12.10247854312365756521334695418, −10.84744850942851096077045853733, −10.27200894548884867423352517650, −9.388145382730066799375716957758, −8.314068662921620801870366634370, −7.32935553924737234554635715193, −6.17607811391304000459219372095, −5.23120826126206387860406263289, −4.03163610530703696568759421429, −2.84808376295958193657944876939,
0.873211372093481929193817551699, 2.11765606463432876756980700565, 3.96386371826481454815970419483, 4.94588483357517826607584749653, 6.43196317598894624171780771890, 7.50561506889397753966929602314, 8.379177967090457726440740527068, 9.378911010948735265793816165876, 10.35907174616548978450295808991, 11.53240124724234424485576391015