L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + 0.999·12-s + 13-s + (−0.499 + 0.866i)16-s + (0.5 + 0.866i)19-s − 0.999·21-s + 0.999·27-s + (0.499 − 0.866i)28-s + 2·31-s + (−0.499 + 0.866i)36-s + (−1 + 1.73i)37-s + (−0.5 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + 0.999·12-s + 13-s + (−0.499 + 0.866i)16-s + (0.5 + 0.866i)19-s − 0.999·21-s + 0.999·27-s + (0.499 − 0.866i)28-s + 2·31-s + (−0.499 + 0.866i)36-s + (−1 + 1.73i)37-s + (−0.5 + 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8215877037\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8215877037\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - 2T + T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−10.18176660991191079611971912361, −9.675788111142042728602167909177, −8.719161452773775644514352834881, −8.220949813341573875702182111727, −6.45918894114801104038988629846, −5.90453157103191901404104566194, −5.08766436698850075897926323509, −4.38066645668035350937579982274, −3.16347999849495124494618806665, −1.41571598164591807696615254989,
1.02971418541924725018879198328, 2.65389537912133398228834367011, 3.91891253518455550192590590877, 4.79094098630065470339421470341, 5.85566278588905642219915718281, 6.99945707754471641375723702135, 7.47628791990059865873293477940, 8.355645739792511022006998320294, 8.972571723690020651103881054325, 10.30453205798940302719406664046