L(s) = 1 | + 0.684·2-s + 3-s − 1.53·4-s − 1.50·5-s + 0.684·6-s + 2.85·7-s − 2.41·8-s + 9-s − 1.03·10-s − 11-s − 1.53·12-s + 0.261·13-s + 1.95·14-s − 1.50·15-s + 1.40·16-s − 3.17·17-s + 0.684·18-s + 1.08·19-s + 2.30·20-s + 2.85·21-s − 0.684·22-s + 7.64·23-s − 2.41·24-s − 2.72·25-s + 0.179·26-s + 27-s − 4.37·28-s + ⋯ |
L(s) = 1 | + 0.484·2-s + 0.577·3-s − 0.765·4-s − 0.674·5-s + 0.279·6-s + 1.08·7-s − 0.854·8-s + 0.333·9-s − 0.326·10-s − 0.301·11-s − 0.442·12-s + 0.0725·13-s + 0.522·14-s − 0.389·15-s + 0.351·16-s − 0.768·17-s + 0.161·18-s + 0.249·19-s + 0.516·20-s + 0.623·21-s − 0.145·22-s + 1.59·23-s − 0.493·24-s − 0.545·25-s + 0.0351·26-s + 0.192·27-s − 0.826·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.157871723\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.157871723\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 0.684T + 2T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 - 0.261T + 13T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 - 7.64T + 23T^{2} \) |
| 29 | \( 1 - 9.16T + 29T^{2} \) |
| 31 | \( 1 - 0.955T + 31T^{2} \) |
| 37 | \( 1 - 0.179T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 3.10T + 59T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 9.97T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.774318962031983665206330935147, −8.566265192615983722330268278798, −7.71912009685357168409065482935, −6.95392168684187093942382414702, −5.73970498065452368331090307660, −4.78568476201633551807673885583, −4.40221910444066009828647368656, −3.42852743684212817874440612706, −2.46691102688749076923934944707, −0.920928182130390499802935265904,
0.920928182130390499802935265904, 2.46691102688749076923934944707, 3.42852743684212817874440612706, 4.40221910444066009828647368656, 4.78568476201633551807673885583, 5.73970498065452368331090307660, 6.95392168684187093942382414702, 7.71912009685357168409065482935, 8.566265192615983722330268278798, 8.774318962031983665206330935147