L(s) = 1 | − 0.471i·2-s − 1.04i·3-s + 1.77·4-s + (0.713 − 2.11i)5-s − 0.490·6-s + 1.17i·7-s − 1.78i·8-s + 1.91·9-s + (−1 − 0.336i)10-s + 0.713·11-s − 1.84i·12-s − 4.10i·13-s + 0.554·14-s + (−2.20 − 0.742i)15-s + 2.71·16-s + 2.55i·17-s + ⋯ |
L(s) = 1 | − 0.333i·2-s − 0.600i·3-s + 0.888·4-s + (0.319 − 0.947i)5-s − 0.200·6-s + 0.444i·7-s − 0.630i·8-s + 0.639·9-s + (−0.316 − 0.106i)10-s + 0.215·11-s − 0.533i·12-s − 1.13i·13-s + 0.148·14-s + (−0.569 − 0.191i)15-s + 0.678·16-s + 0.619i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.531503430\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.531503430\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.713 + 2.11i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.471iT - 2T^{2} \) |
| 3 | \( 1 + 1.04iT - 3T^{2} \) |
| 7 | \( 1 - 1.17iT - 7T^{2} \) |
| 11 | \( 1 - 0.713T + 11T^{2} \) |
| 13 | \( 1 + 4.10iT - 13T^{2} \) |
| 17 | \( 1 - 2.55iT - 17T^{2} \) |
| 23 | \( 1 - 0.607iT - 23T^{2} \) |
| 29 | \( 1 - 0.859T + 29T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 9.38iT - 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 4.98iT - 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 5.18iT - 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 12.3iT - 73T^{2} \) |
| 79 | \( 1 - 5.96T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 16.7iT - 97T^{2} \) |
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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
−8.969550490565918244813110290776, −8.169740090604501594878512164224, −7.53672188965176944791456484184, −6.56382692280126836927321879086, −5.94261537185870986302272970016, −5.05536622183067594480882547447, −3.91744215362673922122818733793, −2.76873810768300137109131327225, −1.77369981477163595671865182535, −0.983614181473396986055024569303,
1.61711170371885654179324319260, 2.62503434664485159513402950379, 3.66410933929767787615637388784, 4.52106541217634180960866776087, 5.62810289502732964472690498457, 6.53298366487694690750461540976, 7.06694597209201673107099416500, 7.57205305999646807032259838706, 8.814153357854531421194946568917, 9.682526135209119923264746971407