| L(s) = 1 | − 2.50·2-s + 4.25·4-s + 3.57·5-s + 1.44·7-s − 5.64·8-s − 8.95·10-s + 5.34·11-s − 5.39·13-s − 3.60·14-s + 5.60·16-s + 5.31·17-s + 2.56·19-s + 15.2·20-s − 13.3·22-s + 6.63·23-s + 7.81·25-s + 13.4·26-s + 6.13·28-s − 5.19·29-s + 4.77·31-s − 2.73·32-s − 13.2·34-s + 5.16·35-s − 3.44·37-s − 6.41·38-s − 20.2·40-s − 8.38·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.12·4-s + 1.60·5-s + 0.545·7-s − 1.99·8-s − 2.83·10-s + 1.61·11-s − 1.49·13-s − 0.964·14-s + 1.40·16-s + 1.28·17-s + 0.588·19-s + 3.40·20-s − 2.85·22-s + 1.38·23-s + 1.56·25-s + 2.64·26-s + 1.16·28-s − 0.964·29-s + 0.856·31-s − 0.483·32-s − 2.27·34-s + 0.872·35-s − 0.565·37-s − 1.04·38-s − 3.19·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.512717793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.512717793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 5 | \( 1 - 3.57T + 5T^{2} \) |
| 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 6.63T + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 - 4.77T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 + 8.38T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 - 0.427T + 47T^{2} \) |
| 53 | \( 1 - 3.73T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 - 6.57T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 0.468T + 73T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 + 0.338T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.466799781468925510887771543496, −7.57586606702447390634025784035, −6.99928884554521079171409530802, −6.42126713879077028321396521489, −5.52438924489373127652540218794, −4.86004272761020815704109843696, −3.30746387436419950670755269033, −2.34681565059287616922474275466, −1.58580311931855120928098842905, −0.979088886182612699500234344006,
0.979088886182612699500234344006, 1.58580311931855120928098842905, 2.34681565059287616922474275466, 3.30746387436419950670755269033, 4.86004272761020815704109843696, 5.52438924489373127652540218794, 6.42126713879077028321396521489, 6.99928884554521079171409530802, 7.57586606702447390634025784035, 8.466799781468925510887771543496
