L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (1.23 − 0.712i)5-s + (0.484 + 2.60i)7-s + 0.999i·8-s + (−0.712 + 1.23i)10-s + (3.13 + 1.09i)11-s + 2.39·13-s + (−1.72 − 2.01i)14-s + (−0.5 − 0.866i)16-s + (0.529 − 0.917i)17-s + (2.37 + 4.11i)19-s − 1.42i·20-s + (−3.25 + 0.620i)22-s + (−1.72 − 2.97i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.551 − 0.318i)5-s + (0.183 + 0.983i)7-s + 0.353i·8-s + (−0.225 + 0.390i)10-s + (0.944 + 0.329i)11-s + 0.663·13-s + (−0.459 − 0.537i)14-s + (−0.125 − 0.216i)16-s + (0.128 − 0.222i)17-s + (0.544 + 0.943i)19-s − 0.318i·20-s + (−0.694 + 0.132i)22-s + (−0.358 − 0.621i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.509024141\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.509024141\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.484 - 2.60i)T \) |
| 11 | \( 1 + (-3.13 - 1.09i)T \) |
good | 5 | \( 1 + (-1.23 + 0.712i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 + (-0.529 + 0.917i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.37 - 4.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.72 + 2.97i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.13iT - 29T^{2} \) |
| 31 | \( 1 + (-0.122 - 0.0706i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.19 - 2.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 - 5.73iT - 43T^{2} \) |
| 47 | \( 1 + (-3.08 + 1.78i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.98 + 3.44i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.71 - 2.72i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.58 - 9.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.567 - 0.983i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + (-0.969 + 1.67i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.82 + 2.20i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.73T + 83T^{2} \) |
| 89 | \( 1 + (-1.67 + 0.969i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 18.9iT - 97T^{2} \) |
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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
−9.704026019359690392602054426019, −8.797640498477056058328894596352, −8.357506432380528587213993529049, −7.33699224708601209041094509145, −6.29349237791171146568860542634, −5.80879491374112579916443377124, −4.88760865050451404761030885597, −3.62976965567867300223610297268, −2.21541026082590409633749032928, −1.28845851526235982401025975555,
0.878591263247198454021777986780, 1.93738959999242685233446802899, 3.32718084273196962439929567216, 4.01860305574477744736488626469, 5.30671329160460620723861663984, 6.44905705467588026011012137105, 6.97001053502146467789004451911, 7.927896485851230677357348958213, 8.757379976980497022769948940027, 9.527623071557617764326597849810