L(s) = 1 | − 2-s − 3-s + 4-s + 2.82·5-s + 6-s − 2.20·7-s − 8-s + 9-s − 2.82·10-s + 11-s − 12-s + 5.81·13-s + 2.20·14-s − 2.82·15-s + 16-s − 0.293·17-s − 18-s − 4.28·19-s + 2.82·20-s + 2.20·21-s − 22-s − 6.10·23-s + 24-s + 3.00·25-s − 5.81·26-s − 27-s − 2.20·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.26·5-s + 0.408·6-s − 0.834·7-s − 0.353·8-s + 0.333·9-s − 0.894·10-s + 0.301·11-s − 0.288·12-s + 1.61·13-s + 0.590·14-s − 0.730·15-s + 0.250·16-s − 0.0710·17-s − 0.235·18-s − 0.982·19-s + 0.632·20-s + 0.481·21-s − 0.213·22-s − 1.27·23-s + 0.204·24-s + 0.601·25-s − 1.13·26-s − 0.192·27-s − 0.417·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 13 | \( 1 - 5.81T + 13T^{2} \) |
| 17 | \( 1 + 0.293T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 - 8.43T + 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 - 2.02T + 59T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 3.15T + 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 + 2.17T + 83T^{2} \) |
| 89 | \( 1 + 3.57T + 89T^{2} \) |
| 97 | \( 1 - 1.22T + 97T^{2} \) |
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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
−8.303149274398729212091953691034, −7.15632527476329096785790231563, −6.48651889569249679549763193633, −5.97677005955119192206383460160, −5.55614101096712258253353557020, −4.16297130642549108238172233413, −3.35975057387512319160515525061, −2.08875453244963999341472415709, −1.43851820951758176166911953819, 0,
1.43851820951758176166911953819, 2.08875453244963999341472415709, 3.35975057387512319160515525061, 4.16297130642549108238172233413, 5.55614101096712258253353557020, 5.97677005955119192206383460160, 6.48651889569249679549763193633, 7.15632527476329096785790231563, 8.303149274398729212091953691034