L(s) = 1 | + (2.23 − 0.152i)5-s + 4.09i·7-s − 0.777·11-s + 1.67i·13-s + 3.55i·17-s + 4.61·19-s − i·23-s + (4.95 − 0.682i)25-s − 8.24·29-s + 6.43·31-s + (0.626 + 9.13i)35-s − 1.05i·37-s + 5.98·41-s − 3.37i·43-s − 1.24i·47-s + ⋯ |
L(s) = 1 | + (0.997 − 0.0684i)5-s + 1.54i·7-s − 0.234·11-s + 0.464i·13-s + 0.862i·17-s + 1.05·19-s − 0.208i·23-s + (0.990 − 0.136i)25-s − 1.53·29-s + 1.15·31-s + (0.105 + 1.54i)35-s − 0.173i·37-s + 0.935·41-s − 0.514i·43-s − 0.181i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0684 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0684 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.163089615\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.163089615\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 + 0.152i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 - 4.09iT - 7T^{2} \) |
| 11 | \( 1 + 0.777T + 11T^{2} \) |
| 13 | \( 1 - 1.67iT - 13T^{2} \) |
| 17 | \( 1 - 3.55iT - 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 + 1.05iT - 37T^{2} \) |
| 41 | \( 1 - 5.98T + 41T^{2} \) |
| 43 | \( 1 + 3.37iT - 43T^{2} \) |
| 47 | \( 1 + 1.24iT - 47T^{2} \) |
| 53 | \( 1 - 6.08iT - 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 + 2.12T + 61T^{2} \) |
| 67 | \( 1 - 14.7iT - 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 15.7iT - 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 + 17.0iT - 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 - 8.78iT - 97T^{2} \) |
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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
−8.869711089441124836045438110890, −7.984084646972609313741299602806, −7.08645868581606164647623450696, −6.14120257224554110662051411843, −5.71278135263167979251473576587, −5.14702269227359957126295595931, −4.09067203243377340317399740317, −2.88737944907081056392015629067, −2.28063920708390588932357040801, −1.37418850765067376382356631678,
0.62038886687056769095146955378, 1.55203230953233857379798430917, 2.76939773434856299919929797451, 3.51498194995924814941390503440, 4.54791892136718935372094023689, 5.23053538043173244005562136687, 6.01946543762877969128497221473, 6.85063343013314087521158826403, 7.48297080847228771352702349042, 7.999550547626362099388427147780