L(s) = 1 | − 0.585·5-s + 5.15·7-s − 11.8·11-s + 14.9i·13-s + 8.91·17-s + (−18.4 + 4.41i)19-s + 35.6·23-s − 24.6·25-s − 7.12i·29-s + 10.8i·31-s − 3.02·35-s − 51.3i·37-s + 33.2i·41-s − 54.2·43-s + 32.6·47-s + ⋯ |
L(s) = 1 | − 0.117·5-s + 0.737·7-s − 1.07·11-s + 1.14i·13-s + 0.524·17-s + (−0.972 + 0.232i)19-s + 1.55·23-s − 0.986·25-s − 0.245i·29-s + 0.348i·31-s − 0.0863·35-s − 1.38i·37-s + 0.810i·41-s − 1.26·43-s + 0.694·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02312271598\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02312271598\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (18.4 - 4.41i)T \) |
good | 5 | \( 1 + 0.585T + 25T^{2} \) |
| 7 | \( 1 - 5.15T + 49T^{2} \) |
| 11 | \( 1 + 11.8T + 121T^{2} \) |
| 13 | \( 1 - 14.9iT - 169T^{2} \) |
| 17 | \( 1 - 8.91T + 289T^{2} \) |
| 23 | \( 1 - 35.6T + 529T^{2} \) |
| 29 | \( 1 + 7.12iT - 841T^{2} \) |
| 31 | \( 1 - 10.8iT - 961T^{2} \) |
| 37 | \( 1 + 51.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 33.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 54.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 32.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 91.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 107. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 23.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 107. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 72.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.25T + 5.32e3T^{2} \) |
| 79 | \( 1 + 45.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 132.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 42.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143. iT - 9.40e3T^{2} \) |
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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
−9.041700075026749023325676367482, −8.180638707312305864150432992000, −7.67380296195995591691002227717, −6.83703298039361314344860958528, −5.96842269562587848113112917290, −5.05007805115498546339719338303, −4.48131504161109559816887180575, −3.45601633351321163415819284471, −2.35798238564336845541091607021, −1.49247768136003596822311782386,
0.00528375849895951542518361141, 1.25107358308354890092901040471, 2.47346403655978105379669260538, 3.24937164674000419034934458919, 4.38128269535087692587985287611, 5.21889527741056974692032147250, 5.68146520040900525094509594659, 6.86488219958845321285764470292, 7.57428532355368898931837809113, 8.297064635088372380581453183749