L(s) = 1 | + 1.58·2-s + 0.511·4-s − 5-s − 2.52·7-s − 2.35·8-s − 1.58·10-s − 4.11·11-s + 5.17·13-s − 3.99·14-s − 4.76·16-s + 0.112·17-s + 3.80·19-s − 0.511·20-s − 6.51·22-s − 1.26·23-s + 25-s + 8.20·26-s − 1.29·28-s − 5.71·29-s + 3.49·31-s − 2.83·32-s + 0.178·34-s + 2.52·35-s + 4.93·37-s + 6.02·38-s + 2.35·40-s + 10.7·41-s + ⋯ |
L(s) = 1 | + 1.12·2-s + 0.255·4-s − 0.447·5-s − 0.953·7-s − 0.833·8-s − 0.501·10-s − 1.24·11-s + 1.43·13-s − 1.06·14-s − 1.19·16-s + 0.0273·17-s + 0.872·19-s − 0.114·20-s − 1.38·22-s − 0.263·23-s + 0.200·25-s + 1.60·26-s − 0.244·28-s − 1.06·29-s + 0.627·31-s − 0.500·32-s + 0.0306·34-s + 0.426·35-s + 0.812·37-s + 0.978·38-s + 0.372·40-s + 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.039455065\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.039455065\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 - 1.58T + 2T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + 4.11T + 11T^{2} \) |
| 13 | \( 1 - 5.17T + 13T^{2} \) |
| 17 | \( 1 - 0.112T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 5.71T + 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 + 9.15T + 47T^{2} \) |
| 53 | \( 1 + 1.29T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 + 7.52T + 61T^{2} \) |
| 67 | \( 1 - 5.42T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 - 9.84T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 97 | \( 1 - 1.86T + 97T^{2} \) |
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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
−8.261763765661095947586332913142, −7.75166423910418823415970677597, −6.70740914049114026947606051945, −6.02909398091849554479067588469, −5.49747439976432867934917563339, −4.62679469622985848728642629794, −3.75484395424243480063500636244, −3.27729735355717584163696334362, −2.43620995238216336203433825923, −0.66964283490668995721787186317,
0.66964283490668995721787186317, 2.43620995238216336203433825923, 3.27729735355717584163696334362, 3.75484395424243480063500636244, 4.62679469622985848728642629794, 5.49747439976432867934917563339, 6.02909398091849554479067588469, 6.70740914049114026947606051945, 7.75166423910418823415970677597, 8.261763765661095947586332913142