| L(s) = 1 | − 1.15·2-s − 2.77·3-s − 0.658·4-s + 2.87·5-s + 3.21·6-s + 3.24·7-s + 3.07·8-s + 4.70·9-s − 3.33·10-s − 2.26·11-s + 1.82·12-s − 1.37·13-s − 3.76·14-s − 7.98·15-s − 2.24·16-s + 8.07·17-s − 5.44·18-s − 1.03·19-s − 1.89·20-s − 9.01·21-s + 2.62·22-s + 6.96·23-s − 8.54·24-s + 3.28·25-s + 1.58·26-s − 4.72·27-s − 2.14·28-s + ⋯ |
| L(s) = 1 | − 0.818·2-s − 1.60·3-s − 0.329·4-s + 1.28·5-s + 1.31·6-s + 1.22·7-s + 1.08·8-s + 1.56·9-s − 1.05·10-s − 0.682·11-s + 0.527·12-s − 0.380·13-s − 1.00·14-s − 2.06·15-s − 0.562·16-s + 1.95·17-s − 1.28·18-s − 0.237·19-s − 0.424·20-s − 1.96·21-s + 0.559·22-s + 1.45·23-s − 1.74·24-s + 0.657·25-s + 0.311·26-s − 0.909·27-s − 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5225406280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5225406280\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.15T + 2T^{2} \) |
| 3 | \( 1 + 2.77T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 - 8.07T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 + 3.54T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 9.25T + 43T^{2} \) |
| 47 | \( 1 - 3.32T + 47T^{2} \) |
| 53 | \( 1 - 3.10T + 53T^{2} \) |
| 59 | \( 1 - 3.63T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 + 5.76T + 83T^{2} \) |
| 89 | \( 1 + 9.29T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
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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
−13.70637706060865846726958892623, −12.63177359566621863701270953465, −11.44279930292446326875447604371, −10.37990876181945059365744297812, −9.941493991262113288772146004551, −8.407085298911093033700204052892, −7.09205025045382319084931393555, −5.39662390128266575840889470387, −5.06934006469129267335816126098, −1.36529778302896490572954442247,
1.36529778302896490572954442247, 5.06934006469129267335816126098, 5.39662390128266575840889470387, 7.09205025045382319084931393555, 8.407085298911093033700204052892, 9.941493991262113288772146004551, 10.37990876181945059365744297812, 11.44279930292446326875447604371, 12.63177359566621863701270953465, 13.70637706060865846726958892623
