L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 12-s + 13-s − 16-s + 2·17-s − 18-s − 4·19-s − 8·23-s + 3·24-s − 26-s + 27-s + 6·29-s − 5·32-s − 2·34-s − 36-s − 2·37-s + 4·38-s + 39-s − 6·41-s + 8·43-s + 8·46-s + 4·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 1.66·23-s + 0.612·24-s − 0.196·26-s + 0.192·27-s + 1.11·29-s − 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.328·37-s + 0.648·38-s + 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.17·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.325049108\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.325049108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.39688076044326, −14.14939009579510, −13.56816830143906, −13.18615734759159, −12.51491360245119, −12.07643512326527, −11.45899175608792, −10.62021433386437, −10.25278312114783, −9.970879673542792, −9.253263896153154, −8.708513156836410, −8.381139928466925, −7.919615696594088, −7.303250053723175, −6.746190463133721, −5.945298900877262, −5.457743911438683, −4.522734077282325, −4.181413606369949, −3.598302176092006, −2.727647218795459, −2.013115811357486, −1.334284205365600, −0.4692855424576425,
0.4692855424576425, 1.334284205365600, 2.013115811357486, 2.727647218795459, 3.598302176092006, 4.181413606369949, 4.522734077282325, 5.457743911438683, 5.945298900877262, 6.746190463133721, 7.303250053723175, 7.919615696594088, 8.381139928466925, 8.708513156836410, 9.253263896153154, 9.970879673542792, 10.25278312114783, 10.62021433386437, 11.45899175608792, 12.07643512326527, 12.51491360245119, 13.18615734759159, 13.56816830143906, 14.14939009579510, 14.39688076044326