L(s) = 1 | + 1.79·2-s + 1.63·3-s + 1.22·4-s + 2.71·5-s + 2.94·6-s + 4.07·7-s − 1.38·8-s − 0.320·9-s + 4.88·10-s − 2.21·11-s + 2.01·12-s + 1.69·13-s + 7.31·14-s + 4.45·15-s − 4.94·16-s + 17-s − 0.576·18-s − 5.88·19-s + 3.34·20-s + 6.66·21-s − 3.97·22-s − 4.49·23-s − 2.26·24-s + 2.39·25-s + 3.03·26-s − 5.43·27-s + 5.00·28-s + ⋯ |
L(s) = 1 | + 1.27·2-s + 0.945·3-s + 0.614·4-s + 1.21·5-s + 1.20·6-s + 1.53·7-s − 0.489·8-s − 0.106·9-s + 1.54·10-s − 0.666·11-s + 0.580·12-s + 0.468·13-s + 1.95·14-s + 1.14·15-s − 1.23·16-s + 0.242·17-s − 0.135·18-s − 1.35·19-s + 0.747·20-s + 1.45·21-s − 0.847·22-s − 0.936·23-s − 0.462·24-s + 0.478·25-s + 0.595·26-s − 1.04·27-s + 0.945·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.841823907\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.841823907\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 3 | \( 1 - 1.63T + 3T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 - 4.07T + 7T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 4.49T + 23T^{2} \) |
| 29 | \( 1 + 8.25T + 29T^{2} \) |
| 31 | \( 1 - 1.95T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 + 6.88T + 43T^{2} \) |
| 47 | \( 1 - 5.32T + 47T^{2} \) |
| 53 | \( 1 - 9.80T + 53T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 1.53T + 83T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 + 5.90T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.942428043448511814129348474303, −8.990475818609631349904263239130, −8.363458528346123236837580983612, −7.54583812731935121261545913237, −6.02806168770564426156234931411, −5.67453384869094901984720879939, −4.65700624770385487583090627620, −3.81615062993883601229190097415, −2.48719910589127961548226228012, −1.96886384709548809447543443827,
1.96886384709548809447543443827, 2.48719910589127961548226228012, 3.81615062993883601229190097415, 4.65700624770385487583090627620, 5.67453384869094901984720879939, 6.02806168770564426156234931411, 7.54583812731935121261545913237, 8.363458528346123236837580983612, 8.990475818609631349904263239130, 9.942428043448511814129348474303