L(s) = 1 | + 2.41i·2-s + 0.537i·3-s − 3.85·4-s + (−2.07 + 0.826i)5-s − 1.29·6-s − 3.18i·7-s − 4.49i·8-s + 2.71·9-s + (−2 − 5.02i)10-s + 4.15·11-s − 2.07i·12-s + 2.07i·13-s + 7.71·14-s + (−0.443 − 1.11i)15-s + 3.15·16-s + 5.79i·17-s + ⋯ |
L(s) = 1 | + 1.71i·2-s + 0.310i·3-s − 1.92·4-s + (−0.929 + 0.369i)5-s − 0.530·6-s − 1.20i·7-s − 1.58i·8-s + 0.903·9-s + (−0.632 − 1.58i)10-s + 1.25·11-s − 0.597i·12-s + 0.574i·13-s + 2.06·14-s + (−0.114 − 0.288i)15-s + 0.788·16-s + 1.40i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.287027449\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.287027449\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.07 - 0.826i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.41iT - 2T^{2} \) |
| 3 | \( 1 - 0.537iT - 3T^{2} \) |
| 7 | \( 1 + 3.18iT - 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.07iT - 13T^{2} \) |
| 17 | \( 1 - 5.79iT - 17T^{2} \) |
| 23 | \( 1 - 2.60iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2.59T + 31T^{2} \) |
| 37 | \( 1 + 4.30iT - 37T^{2} \) |
| 41 | \( 1 - 0.599T + 41T^{2} \) |
| 43 | \( 1 + 3.18iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 11.7iT - 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 + 8.75T + 61T^{2} \) |
| 67 | \( 1 + 4.76iT - 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 2.72iT - 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 - 7.07iT - 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 2.07iT - 97T^{2} \) |
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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
−9.363023213588967663552981567946, −8.766938755025454326618737491434, −7.70349621175330964154329825677, −7.40460647885912661252227599668, −6.62924133858571111476306348373, −6.11066428776698731758569280567, −4.61693235331069114780833830615, −4.19036253732449386584103905424, −3.64631606416388633308605903292, −1.23957675572701137288802427525,
0.58369764332420719095883884932, 1.61099698428263238427734397148, 2.72044275827066815376355793963, 3.53230361095427239104790116697, 4.48504821491565818703401380171, 5.08129568667083050599157896054, 6.47157302246313282062509013641, 7.38383122863973894201064014406, 8.501320771828631491737294669643, 8.947155999282684307226545114000