L(s) = 1 | − 0.924·2-s − 3.36·3-s − 1.14·4-s + 3.11·6-s − 2.46·7-s + 2.90·8-s + 8.31·9-s − 3.35·11-s + 3.85·12-s + 3.13·13-s + 2.27·14-s − 0.397·16-s − 6.74·17-s − 7.68·18-s − 6.04·19-s + 8.29·21-s + 3.10·22-s + 1.93·23-s − 9.78·24-s − 2.90·26-s − 17.8·27-s + 2.82·28-s − 4.35·29-s − 9.83·31-s − 5.44·32-s + 11.2·33-s + 6.23·34-s + ⋯ |
L(s) = 1 | − 0.653·2-s − 1.94·3-s − 0.572·4-s + 1.26·6-s − 0.931·7-s + 1.02·8-s + 2.77·9-s − 1.01·11-s + 1.11·12-s + 0.870·13-s + 0.608·14-s − 0.0993·16-s − 1.63·17-s − 1.81·18-s − 1.38·19-s + 1.80·21-s + 0.661·22-s + 0.402·23-s − 1.99·24-s − 0.569·26-s − 3.44·27-s + 0.533·28-s − 0.808·29-s − 1.76·31-s − 0.963·32-s + 1.96·33-s + 1.06·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09797476394\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09797476394\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 + 0.924T + 2T^{2} \) |
| 3 | \( 1 + 3.36T + 3T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 3.13T + 13T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 + 6.04T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 + 9.83T + 31T^{2} \) |
| 37 | \( 1 - 2.88T + 37T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 + 1.83T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 1.31T + 53T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 - 5.81T + 67T^{2} \) |
| 71 | \( 1 - 1.34T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 - 9.96T + 79T^{2} \) |
| 83 | \( 1 - 8.66T + 83T^{2} \) |
| 89 | \( 1 + 1.18T + 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−9.590492534725152444302955132949, −8.932399730863454472618003667226, −7.83255069045549144856150391436, −6.86509630866156017710770015533, −6.29274939568927047358899607039, −5.42848759185174753931678511644, −4.62731293219311450456268841370, −3.82711263025172539841517127936, −1.81079340503616322714293467876, −0.26714978711785498054195124139,
0.26714978711785498054195124139, 1.81079340503616322714293467876, 3.82711263025172539841517127936, 4.62731293219311450456268841370, 5.42848759185174753931678511644, 6.29274939568927047358899607039, 6.86509630866156017710770015533, 7.83255069045549144856150391436, 8.932399730863454472618003667226, 9.590492534725152444302955132949