L(s) = 1 | + 2.45i·2-s − 1.56i·3-s − 4.03·4-s + (2.19 + 0.407i)5-s + 3.83·6-s + 4.50i·7-s − 4.99i·8-s + 0.563·9-s + (−1 + 5.40i)10-s + 2.19·11-s + 6.29i·12-s + 3.75i·13-s − 11.0·14-s + (0.635 − 3.43i)15-s + 4.19·16-s + 0.665i·17-s + ⋯ |
L(s) = 1 | + 1.73i·2-s − 0.901i·3-s − 2.01·4-s + (0.983 + 0.182i)5-s + 1.56·6-s + 1.70i·7-s − 1.76i·8-s + 0.187·9-s + (−0.316 + 1.70i)10-s + 0.662·11-s + 1.81i·12-s + 1.04i·13-s − 2.95·14-s + (0.164 − 0.886i)15-s + 1.04·16-s + 0.161i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.902338285\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.902338285\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.19 - 0.407i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.45iT - 2T^{2} \) |
| 3 | \( 1 + 1.56iT - 3T^{2} \) |
| 7 | \( 1 - 4.50iT - 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 - 3.75iT - 13T^{2} \) |
| 17 | \( 1 - 0.665iT - 17T^{2} \) |
| 23 | \( 1 - 0.488iT - 23T^{2} \) |
| 29 | \( 1 + 3.59T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 + 3.01iT - 37T^{2} \) |
| 41 | \( 1 - 0.0724T + 41T^{2} \) |
| 43 | \( 1 + 0.420iT - 43T^{2} \) |
| 47 | \( 1 - 5.02iT - 47T^{2} \) |
| 53 | \( 1 + 2.61iT - 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 67 | \( 1 - 5.72iT - 67T^{2} \) |
| 71 | \( 1 - 6.97T + 71T^{2} \) |
| 73 | \( 1 - 2.95iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 15.6iT - 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 - 4.38iT - 97T^{2} \) |
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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
−9.297205106258142745452560872701, −8.703239351756146430249214680814, −7.967009653954629074731858803726, −6.95884128388815762836321557968, −6.49629528574991758505944069674, −5.95022452421639469327968403691, −5.26813141298102754089395185362, −4.25324810995822957970619935575, −2.53173245670575417876754799765, −1.57972610814576505546006099158,
0.76930858935707500919458815039, 1.61551352178226160454637382829, 2.99371694333522433594829729451, 3.74107137521567807169881598465, 4.47509036754852416689373231080, 5.08200996611406346461740704427, 6.38237934051852510791521689550, 7.44783249705816940480444050936, 8.595231661229073478398877986270, 9.465172832393953573174347465085