L(s) = 1 | − 1.31·2-s − 0.276·4-s + 3.45·5-s − 7-s + 2.98·8-s − 4.54·10-s + 5.53·13-s + 1.31·14-s − 3.37·16-s + 4.96·17-s − 8.22·19-s − 0.955·20-s + 7.56·23-s + 6.96·25-s − 7.27·26-s + 0.276·28-s − 1.47·29-s + 2.41·31-s − 1.55·32-s − 6.51·34-s − 3.45·35-s − 2.83·37-s + 10.8·38-s + 10.3·40-s + 6.93·41-s − 5.62·43-s − 9.92·46-s + ⋯ |
L(s) = 1 | − 0.928·2-s − 0.138·4-s + 1.54·5-s − 0.377·7-s + 1.05·8-s − 1.43·10-s + 1.53·13-s + 0.350·14-s − 0.842·16-s + 1.20·17-s − 1.88·19-s − 0.213·20-s + 1.57·23-s + 1.39·25-s − 1.42·26-s + 0.0522·28-s − 0.272·29-s + 0.433·31-s − 0.274·32-s − 1.11·34-s − 0.584·35-s − 0.466·37-s + 1.75·38-s + 1.63·40-s + 1.08·41-s − 0.858·43-s − 1.46·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.721479205\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.721479205\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 5 | \( 1 - 3.45T + 5T^{2} \) |
| 13 | \( 1 - 5.53T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 + 8.22T + 19T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 + 5.62T + 43T^{2} \) |
| 47 | \( 1 - 0.0417T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 - 2.99T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 + 7.38T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 9.64T + 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 + 2.35T + 97T^{2} \) |
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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
−8.228962149156793701493495684684, −7.12032329512673300201763580721, −6.55682427552178822566661954847, −5.84279717311294733397560525016, −5.29005199363465800925661472042, −4.32134000451620516796221250343, −3.45356032442340594130384944554, −2.40345236211070671232720117055, −1.53633250786120851048187376996, −0.833526768094489858765390349809,
0.833526768094489858765390349809, 1.53633250786120851048187376996, 2.40345236211070671232720117055, 3.45356032442340594130384944554, 4.32134000451620516796221250343, 5.29005199363465800925661472042, 5.84279717311294733397560525016, 6.55682427552178822566661954847, 7.12032329512673300201763580721, 8.228962149156793701493495684684