L(s) = 1 | − 2.34·2-s + 3.49·4-s − 3.40·5-s − 2.95·7-s − 3.50·8-s + 7.97·10-s − 11-s + 5.60·13-s + 6.93·14-s + 1.23·16-s + 3.44·17-s + 6.93·19-s − 11.8·20-s + 2.34·22-s − 0.836·23-s + 6.57·25-s − 13.1·26-s − 10.3·28-s − 6.68·29-s + 0.117·31-s + 4.12·32-s − 8.06·34-s + 10.0·35-s − 0.152·37-s − 16.2·38-s + 11.9·40-s + 10.6·41-s + ⋯ |
L(s) = 1 | − 1.65·2-s + 1.74·4-s − 1.52·5-s − 1.11·7-s − 1.24·8-s + 2.52·10-s − 0.301·11-s + 1.55·13-s + 1.85·14-s + 0.307·16-s + 0.834·17-s + 1.59·19-s − 2.65·20-s + 0.499·22-s − 0.174·23-s + 1.31·25-s − 2.57·26-s − 1.95·28-s − 1.24·29-s + 0.0210·31-s + 0.729·32-s − 1.38·34-s + 1.70·35-s − 0.0250·37-s − 2.63·38-s + 1.88·40-s + 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4766784966\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4766784966\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 + 2.95T + 7T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 6.93T + 19T^{2} \) |
| 23 | \( 1 + 0.836T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 0.117T + 31T^{2} \) |
| 37 | \( 1 + 0.152T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 + 0.889T + 47T^{2} \) |
| 53 | \( 1 + 3.79T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 67 | \( 1 - 2.66T + 67T^{2} \) |
| 71 | \( 1 + 1.13T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 2.71T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−7.971551592064745040591862931467, −7.65391501146611511270239949739, −7.06990466945476431027034155115, −6.23305926118158289998042687388, −5.48120931433481931952382179396, −4.07467097462352269209128262697, −3.49226075868698767195524200594, −2.78313489646529542883149903226, −1.28931020500917756085555491963, −0.52789755650064710048276365649,
0.52789755650064710048276365649, 1.28931020500917756085555491963, 2.78313489646529542883149903226, 3.49226075868698767195524200594, 4.07467097462352269209128262697, 5.48120931433481931952382179396, 6.23305926118158289998042687388, 7.06990466945476431027034155115, 7.65391501146611511270239949739, 7.971551592064745040591862931467