L(s) = 1 | − 2.65·2-s + 2.58·3-s + 5.06·4-s + 1.09·5-s − 6.86·6-s + 7-s − 8.13·8-s + 3.66·9-s − 2.91·10-s − 0.201·11-s + 13.0·12-s + 6.89·13-s − 2.65·14-s + 2.83·15-s + 11.4·16-s − 3.73·17-s − 9.75·18-s − 0.944·19-s + 5.55·20-s + 2.58·21-s + 0.536·22-s + 9.13·23-s − 21.0·24-s − 3.79·25-s − 18.3·26-s + 1.72·27-s + 5.06·28-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 1.49·3-s + 2.53·4-s + 0.490·5-s − 2.80·6-s + 0.377·7-s − 2.87·8-s + 1.22·9-s − 0.922·10-s − 0.0608·11-s + 3.77·12-s + 1.91·13-s − 0.710·14-s + 0.732·15-s + 2.87·16-s − 0.905·17-s − 2.29·18-s − 0.216·19-s + 1.24·20-s + 0.563·21-s + 0.114·22-s + 1.90·23-s − 4.28·24-s − 0.758·25-s − 3.59·26-s + 0.332·27-s + 0.956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.960979157\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.960979157\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 - 2.58T + 3T^{2} \) |
| 5 | \( 1 - 1.09T + 5T^{2} \) |
| 11 | \( 1 + 0.201T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 + 0.944T + 19T^{2} \) |
| 23 | \( 1 - 9.13T + 23T^{2} \) |
| 29 | \( 1 - 1.11T + 29T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 + 0.763T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 - 2.11T + 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 4.53T + 89T^{2} \) |
| 97 | \( 1 - 3.12T + 97T^{2} \) |
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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
−8.411479203058553597735079636609, −7.70039750872057731273226546683, −7.00467646576629383231096509700, −6.39118982456327493954792605538, −5.50089726916242675091813922035, −4.05058463038179126470100356734, −3.22136206864763160686007926968, −2.40414555034135578343324494415, −1.76228821449357595522476436090, −0.956476739963700331957974865385,
0.956476739963700331957974865385, 1.76228821449357595522476436090, 2.40414555034135578343324494415, 3.22136206864763160686007926968, 4.05058463038179126470100356734, 5.50089726916242675091813922035, 6.39118982456327493954792605538, 7.00467646576629383231096509700, 7.70039750872057731273226546683, 8.411479203058553597735079636609