| L(s) = 1 | + 0.0222·2-s + 2.31·3-s − 1.99·4-s − 0.574·5-s + 0.0514·6-s − 0.0888·8-s + 2.36·9-s − 0.0127·10-s − 2.61·11-s − 4.63·12-s − 1.33·15-s + 3.99·16-s − 2.92·17-s + 0.0525·18-s − 4.74·19-s + 1.14·20-s − 0.0581·22-s − 6.27·23-s − 0.205·24-s − 4.67·25-s − 1.46·27-s − 3.03·29-s − 0.0295·30-s + 5.84·31-s + 0.266·32-s − 6.06·33-s − 0.0649·34-s + ⋯ |
| L(s) = 1 | + 0.0157·2-s + 1.33·3-s − 0.999·4-s − 0.256·5-s + 0.0210·6-s − 0.0313·8-s + 0.788·9-s − 0.00403·10-s − 0.789·11-s − 1.33·12-s − 0.343·15-s + 0.999·16-s − 0.709·17-s + 0.0123·18-s − 1.08·19-s + 0.256·20-s − 0.0123·22-s − 1.30·23-s − 0.0419·24-s − 0.934·25-s − 0.282·27-s − 0.564·29-s − 0.00539·30-s + 1.04·31-s + 0.0470·32-s − 1.05·33-s − 0.0111·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.687041884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.687041884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.0222T + 2T^{2} \) |
| 3 | \( 1 - 2.31T + 3T^{2} \) |
| 5 | \( 1 + 0.574T + 5T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 17 | \( 1 + 2.92T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 6.27T + 23T^{2} \) |
| 29 | \( 1 + 3.03T + 29T^{2} \) |
| 31 | \( 1 - 5.84T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 9.14T + 41T^{2} \) |
| 43 | \( 1 - 6.22T + 43T^{2} \) |
| 47 | \( 1 + 6.33T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 + 3.52T + 59T^{2} \) |
| 61 | \( 1 - 4.66T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 - 7.47T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + 4.37T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 + 1.69T + 97T^{2} \) |
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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
−7.908692374681790184320748667841, −7.63319639822780084199415893437, −6.35295576523759818049544378273, −5.75891604865512445055220801286, −4.72078426731517055613283808191, −4.10436613410774972430904775586, −3.67307997822423569191299936190, −2.55701047457589533008270997809, −2.12520149530975671949681404530, −0.57259533086272856645893190603,
0.57259533086272856645893190603, 2.12520149530975671949681404530, 2.55701047457589533008270997809, 3.67307997822423569191299936190, 4.10436613410774972430904775586, 4.72078426731517055613283808191, 5.75891604865512445055220801286, 6.35295576523759818049544378273, 7.63319639822780084199415893437, 7.908692374681790184320748667841
