L(s) = 1 | + 1.71·3-s − 3.66·5-s + 2.65i·7-s − 0.0489·9-s + i·11-s − 4.00i·13-s − 6.30·15-s + 5.00·17-s + (1.66 + 4.02i)19-s + 4.55i·21-s + 0.204i·23-s + 8.45·25-s − 5.23·27-s + 6.62i·29-s − 8.91·31-s + ⋯ |
L(s) = 1 | + 0.991·3-s − 1.64·5-s + 1.00i·7-s − 0.0163·9-s + 0.301i·11-s − 1.11i·13-s − 1.62·15-s + 1.21·17-s + (0.383 + 0.923i)19-s + 0.993i·21-s + 0.0425i·23-s + 1.69·25-s − 1.00·27-s + 1.23i·29-s − 1.60·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01122773885\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01122773885\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 19 | \( 1 + (-1.66 - 4.02i)T \) |
good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 + 3.66T + 5T^{2} \) |
| 7 | \( 1 - 2.65iT - 7T^{2} \) |
| 13 | \( 1 + 4.00iT - 13T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 23 | \( 1 - 0.204iT - 23T^{2} \) |
| 29 | \( 1 - 6.62iT - 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 + 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 1.16iT - 41T^{2} \) |
| 43 | \( 1 - 8.59iT - 43T^{2} \) |
| 47 | \( 1 + 7.02iT - 47T^{2} \) |
| 53 | \( 1 + 5.64iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 + 9.02T + 67T^{2} \) |
| 71 | \( 1 - 3.63T + 71T^{2} \) |
| 73 | \( 1 + 7.45T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 3.09iT - 83T^{2} \) |
| 89 | \( 1 - 1.40iT - 89T^{2} \) |
| 97 | \( 1 + 16.0iT - 97T^{2} \) |
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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
−8.846027191081382307382973769908, −8.265245620570352361205009383756, −7.58607890148679635382270039357, −7.37891237491518840554657586158, −5.81054980795375919162101591487, −5.34779955300519643860607409158, −4.14164361305003462131051174610, −3.26714284170144405794223125828, −3.06336906537750828843828197299, −1.67337981603753411962639597895,
0.00306455296322583060954012314, 1.32353888178430188110980659684, 2.83097766704246716779217198106, 3.48093389239663554455497656150, 4.10524522482105860728243253208, 4.77374271711595083045613969139, 6.06974657408093658609002190156, 7.15620794252453382931315409227, 7.57605211417141479382293674542, 8.026503531242010016267865839334