L(s) = 1 | + (0.0777 − 1.41i)2-s + (−1.98 − 0.219i)4-s − 0.438i·5-s + (−0.464 + 2.79i)8-s + (−0.619 − 0.0341i)10-s − 2.11i·11-s + 3.84i·13-s + (3.90 + 0.872i)16-s − 5.64i·17-s − 2.97·19-s + (−0.0963 + 0.872i)20-s + (−2.98 − 0.164i)22-s − 4.77i·23-s + 4.80·25-s + (5.43 + 0.299i)26-s + ⋯ |
L(s) = 1 | + (0.0549 − 0.998i)2-s + (−0.993 − 0.109i)4-s − 0.196i·5-s + (−0.164 + 0.986i)8-s + (−0.196 − 0.0107i)10-s − 0.637i·11-s + 1.06i·13-s + (0.975 + 0.218i)16-s − 1.36i·17-s − 0.682·19-s + (−0.0215 + 0.195i)20-s + (−0.637 − 0.0350i)22-s − 0.994i·23-s + 0.961·25-s + (1.06 + 0.0586i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4619988317\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4619988317\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0777 + 1.41i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.438iT - 5T^{2} \) |
| 11 | \( 1 + 2.11iT - 11T^{2} \) |
| 13 | \( 1 - 3.84iT - 13T^{2} \) |
| 17 | \( 1 + 5.64iT - 17T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 + 4.77iT - 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + 7.42T + 31T^{2} \) |
| 37 | \( 1 + 5.28T + 37T^{2} \) |
| 41 | \( 1 - 6.81iT - 41T^{2} \) |
| 43 | \( 1 + 4.38iT - 43T^{2} \) |
| 47 | \( 1 - 1.68T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 8.11T + 59T^{2} \) |
| 61 | \( 1 - 6.18iT - 61T^{2} \) |
| 67 | \( 1 + 7.85iT - 67T^{2} \) |
| 71 | \( 1 + 1.16iT - 71T^{2} \) |
| 73 | \( 1 - 10.0iT - 73T^{2} \) |
| 79 | \( 1 + 15.5iT - 79T^{2} \) |
| 83 | \( 1 - 5.49T + 83T^{2} \) |
| 89 | \( 1 - 10.4iT - 89T^{2} \) |
| 97 | \( 1 + 2.22iT - 97T^{2} \) |
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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
−9.149000455554762374086287916236, −8.310365895425665338889449479074, −7.26914011689683342834163744644, −6.31292283988247300861086769156, −5.24167855536837943984308559875, −4.55999159987294663139580730825, −3.62296882261664716608513027727, −2.65127594981485515581615142920, −1.60129055609798125631774415925, −0.16881553361392887664000249791,
1.68575329317485246365903913710, 3.30876120807947010944703538173, 4.05575524633514783781943752900, 5.16079580537431330549256847188, 5.77465886869267029903028012860, 6.64819198476270151144249467147, 7.45435943264532877501184242605, 8.031938969461209635667755379271, 8.896683375638672960862050028973, 9.575845855103511678094855727796