| L(s) = 1 | + 0.171·2-s − 1.12·3-s − 1.97·4-s − 2.82·5-s − 0.192·6-s − 2.92·7-s − 0.679·8-s − 1.73·9-s − 0.483·10-s − 3.99·11-s + 2.21·12-s + 2.81·13-s − 0.500·14-s + 3.17·15-s + 3.82·16-s + 0.482·17-s − 0.296·18-s − 3.04·19-s + 5.57·20-s + 3.28·21-s − 0.682·22-s + 5.51·23-s + 0.763·24-s + 2.99·25-s + 0.480·26-s + 5.32·27-s + 5.76·28-s + ⋯ |
| L(s) = 1 | + 0.120·2-s − 0.649·3-s − 0.985·4-s − 1.26·5-s − 0.0784·6-s − 1.10·7-s − 0.240·8-s − 0.578·9-s − 0.152·10-s − 1.20·11-s + 0.639·12-s + 0.779·13-s − 0.133·14-s + 0.820·15-s + 0.956·16-s + 0.117·17-s − 0.0699·18-s − 0.699·19-s + 1.24·20-s + 0.717·21-s − 0.145·22-s + 1.14·23-s + 0.155·24-s + 0.598·25-s + 0.0942·26-s + 1.02·27-s + 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2630973139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2630973139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 0.171T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 - 0.482T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 5.10T + 41T^{2} \) |
| 43 | \( 1 - 8.21T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + 0.446T + 53T^{2} \) |
| 59 | \( 1 - 1.24T + 59T^{2} \) |
| 61 | \( 1 + 8.59T + 61T^{2} \) |
| 67 | \( 1 + 0.944T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 - 0.483T + 73T^{2} \) |
| 79 | \( 1 - 1.85T + 79T^{2} \) |
| 83 | \( 1 - 4.35T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−10.47905412121954092065970427304, −9.192249751753887083184060426213, −8.562178649225386819996564275847, −7.74294946136607039435436977133, −6.66888815720432359000066946431, −5.65733387505460032816084082833, −4.90722893189871953068718608086, −3.76834889048235215287294082997, −3.07870985183737047725896424491, −0.40020865405981684223286852076,
0.40020865405981684223286852076, 3.07870985183737047725896424491, 3.76834889048235215287294082997, 4.90722893189871953068718608086, 5.65733387505460032816084082833, 6.66888815720432359000066946431, 7.74294946136607039435436977133, 8.562178649225386819996564275847, 9.192249751753887083184060426213, 10.47905412121954092065970427304
