L(s) = 1 | + 1.48·2-s + 3-s + 0.210·4-s − 0.524·5-s + 1.48·6-s − 7-s − 2.66·8-s + 9-s − 0.780·10-s + 1.53·11-s + 0.210·12-s + 2.28·13-s − 1.48·14-s − 0.524·15-s − 4.37·16-s − 1.54·17-s + 1.48·18-s + 8.08·19-s − 0.110·20-s − 21-s + 2.28·22-s + 1.77·23-s − 2.66·24-s − 4.72·25-s + 3.40·26-s + 27-s − 0.210·28-s + ⋯ |
L(s) = 1 | + 1.05·2-s + 0.577·3-s + 0.105·4-s − 0.234·5-s + 0.607·6-s − 0.377·7-s − 0.940·8-s + 0.333·9-s − 0.246·10-s + 0.462·11-s + 0.0608·12-s + 0.634·13-s − 0.397·14-s − 0.135·15-s − 1.09·16-s − 0.375·17-s + 0.350·18-s + 1.85·19-s − 0.0247·20-s − 0.218·21-s + 0.486·22-s + 0.369·23-s − 0.543·24-s − 0.944·25-s + 0.667·26-s + 0.192·27-s − 0.0398·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.665521267\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.665521267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 5 | \( 1 + 0.524T + 5T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 + 1.54T + 17T^{2} \) |
| 19 | \( 1 - 8.08T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 - 2.13T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 - 3.15T + 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 - 6.63T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 + 3.43T + 97T^{2} \) |
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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
−7.78019671061976281524370268073, −6.97014611962098601534533982487, −6.44002090951551231260603247175, −5.48751366505086299034992329569, −5.10821853402152408704858408484, −3.99941654701768080237366544845, −3.66089036797009693806352259170, −3.04710243194335959518347058229, −2.04731435980800511932475370993, −0.78742253839482228743367044484,
0.78742253839482228743367044484, 2.04731435980800511932475370993, 3.04710243194335959518347058229, 3.66089036797009693806352259170, 3.99941654701768080237366544845, 5.10821853402152408704858408484, 5.48751366505086299034992329569, 6.44002090951551231260603247175, 6.97014611962098601534533982487, 7.78019671061976281524370268073