L(s) = 1 | + (−2.17 − 1.25i)2-s + (2.16 + 3.75i)4-s + (1.11 + 1.93i)5-s + (−1.31 − 2.29i)7-s − 5.87i·8-s + (0.00343 − 5.62i)10-s + (0.489 + 0.847i)11-s + 5.14i·13-s + (−0.0275 + 6.65i)14-s + (−3.05 + 5.29i)16-s + (−3.59 + 2.07i)17-s + (−1.15 + 1.99i)19-s + (−4.84 + 8.38i)20-s − 2.46i·22-s + (4.39 + 2.53i)23-s + ⋯ |
L(s) = 1 | + (−1.54 − 0.889i)2-s + (1.08 + 1.87i)4-s + (0.499 + 0.866i)5-s + (−0.496 − 0.868i)7-s − 2.07i·8-s + (0.00108 − 1.77i)10-s + (0.147 + 0.255i)11-s + 1.42i·13-s + (−0.00735 + 1.77i)14-s + (−0.763 + 1.32i)16-s + (−0.871 + 0.503i)17-s + (−0.263 + 0.456i)19-s + (−1.08 + 1.87i)20-s − 0.524i·22-s + (0.916 + 0.528i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.533306 + 0.159793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.533306 + 0.159793i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.11 - 1.93i)T \) |
| 7 | \( 1 + (1.31 + 2.29i)T \) |
good | 2 | \( 1 + (2.17 + 1.25i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-0.489 - 0.847i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.14iT - 13T^{2} \) |
| 17 | \( 1 + (3.59 - 2.07i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.15 - 1.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.39 - 2.53i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.92T + 29T^{2} \) |
| 31 | \( 1 + (0.316 + 0.548i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.84 - 4.52i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 - 0.344iT - 43T^{2} \) |
| 47 | \( 1 + (-3.67 - 2.11i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.61 - 3.81i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.908 + 1.57i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.328 - 0.568i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.01 + 4.62i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + (-4.65 + 2.68i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.44 - 9.42i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.62iT - 83T^{2} \) |
| 89 | \( 1 + (8.15 - 14.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.53iT - 97T^{2} \) |
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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
−11.19365295561391041911435679877, −10.80816492701971025414526139685, −9.776643429735399372587174518317, −9.369249676181910523861179458183, −8.164256978054211939009167937952, −7.01181365775099629491243615338, −6.51405645157487120456195417251, −4.13763033851082916894387308812, −2.85123534687918338193757870925, −1.58798980634677938566106701628,
0.68165144515960499668575755966, 2.52633009751200721198589446096, 5.03914203386852333593478119739, 5.94903933729235426549582387949, 6.79113164874953443180437052773, 8.106877363012522177683815229692, 8.755785834773687071706140845713, 9.379045204460153449522687274541, 10.23039195720607486908974188133, 11.20827499737760099121287154372