L(s) = 1 | − 4.47i·3-s − 5i·5-s + 31.3·7-s + 6.99·9-s − 8.94i·11-s + 62i·13-s − 22.3·15-s − 46·17-s − 107. i·19-s − 140i·21-s − 192.·23-s − 25·25-s − 152. i·27-s + 90i·29-s − 152.·31-s + ⋯ |
L(s) = 1 | − 0.860i·3-s − 0.447i·5-s + 1.69·7-s + 0.259·9-s − 0.245i·11-s + 1.32i·13-s − 0.384·15-s − 0.656·17-s − 1.29i·19-s − 1.45i·21-s − 1.74·23-s − 0.200·25-s − 1.08i·27-s + 0.576i·29-s − 0.880·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.155891258\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.155891258\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 5iT \) |
good | 3 | \( 1 + 4.47iT - 27T^{2} \) |
| 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 + 8.94iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 62iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 46T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 90iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 10T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 678iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 411. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 250iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 49.1iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 522T + 3.89e5T^{2} \) |
| 79 | \( 1 - 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 380. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 970T + 7.04e5T^{2} \) |
| 97 | \( 1 + 934T + 9.12e5T^{2} \) |
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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
−8.780416097929618146626344343975, −8.186822949506990530484028470179, −7.33950219276853145308792160334, −6.73554672920332915683664715527, −5.60618084651259818025515095374, −4.64940045137544488462725649025, −4.05262421936386100360940474467, −2.05826005077091904047821762205, −1.79226970452170293497860322022, −0.48551015029238458257720659547,
1.34829455917969117263002784972, 2.38715769650539617571227666553, 3.79535420516868782820266454424, 4.37893453975748396549307363199, 5.31999676510117147675401889573, 6.04600888502554303506236424203, 7.49490885249302586412625899506, 7.891922882013459585771966553157, 8.739457140757484667286748448736, 9.836298123709377668574464757430