L(s) = 1 | − 2-s − 2.89·3-s + 4-s + 5-s + 2.89·6-s + 4.20·7-s − 8-s + 5.36·9-s − 10-s + 4.26·11-s − 2.89·12-s + 0.906·13-s − 4.20·14-s − 2.89·15-s + 16-s − 2.67·17-s − 5.36·18-s − 2.08·19-s + 20-s − 12.1·21-s − 4.26·22-s + 1.19·23-s + 2.89·24-s + 25-s − 0.906·26-s − 6.83·27-s + 4.20·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.66·3-s + 0.5·4-s + 0.447·5-s + 1.18·6-s + 1.59·7-s − 0.353·8-s + 1.78·9-s − 0.316·10-s + 1.28·11-s − 0.834·12-s + 0.251·13-s − 1.12·14-s − 0.746·15-s + 0.250·16-s − 0.648·17-s − 1.26·18-s − 0.477·19-s + 0.223·20-s − 2.65·21-s − 0.909·22-s + 0.248·23-s + 0.590·24-s + 0.200·25-s − 0.177·26-s − 1.31·27-s + 0.795·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.259828460\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.259828460\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 7 | \( 1 - 4.20T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 - 0.906T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 - 3.06T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 - 6.95T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 + 1.32T + 47T^{2} \) |
| 53 | \( 1 - 7.66T + 53T^{2} \) |
| 59 | \( 1 + 2.85T + 59T^{2} \) |
| 61 | \( 1 - 5.70T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 5.58T + 71T^{2} \) |
| 73 | \( 1 + 4.79T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 6.81T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.157227525695688661731969617193, −7.19924698130045036842801339297, −6.55917251966860883961546822069, −6.12486065952486959900741983974, −5.26251433039158311981189895093, −4.66403864584987182627023286606, −3.97252443656825192293255915244, −2.30831491792278397004471950652, −1.42554174792056779934211465033, −0.818824411394994481565096421131,
0.818824411394994481565096421131, 1.42554174792056779934211465033, 2.30831491792278397004471950652, 3.97252443656825192293255915244, 4.66403864584987182627023286606, 5.26251433039158311981189895093, 6.12486065952486959900741983974, 6.55917251966860883961546822069, 7.19924698130045036842801339297, 8.157227525695688661731969617193