| L(s) = 1 | − 2.43·2-s − 3-s + 3.91·4-s + 2.62·5-s + 2.43·6-s − 7-s − 4.66·8-s + 9-s − 6.38·10-s + 1.51·11-s − 3.91·12-s + 1.98·13-s + 2.43·14-s − 2.62·15-s + 3.51·16-s + 7.90·17-s − 2.43·18-s + 4.57·19-s + 10.2·20-s + 21-s − 3.69·22-s − 0.139·23-s + 4.66·24-s + 1.89·25-s − 4.82·26-s − 27-s − 3.91·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.577·3-s + 1.95·4-s + 1.17·5-s + 0.993·6-s − 0.377·7-s − 1.64·8-s + 0.333·9-s − 2.01·10-s + 0.457·11-s − 1.13·12-s + 0.550·13-s + 0.650·14-s − 0.677·15-s + 0.878·16-s + 1.91·17-s − 0.573·18-s + 1.04·19-s + 2.29·20-s + 0.218·21-s − 0.787·22-s − 0.0290·23-s + 0.952·24-s + 0.378·25-s − 0.947·26-s − 0.192·27-s − 0.740·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050092911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050092911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 - 7.90T + 17T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 + 0.139T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 - 9.18T + 31T^{2} \) |
| 37 | \( 1 + 4.66T + 37T^{2} \) |
| 41 | \( 1 + 4.07T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 1.84T + 47T^{2} \) |
| 53 | \( 1 + 6.21T + 53T^{2} \) |
| 59 | \( 1 + 9.34T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 1.59T + 67T^{2} \) |
| 71 | \( 1 - 1.24T + 71T^{2} \) |
| 73 | \( 1 - 5.03T + 73T^{2} \) |
| 79 | \( 1 + 0.556T + 79T^{2} \) |
| 83 | \( 1 + 9.10T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 8.59T + 97T^{2} \) |
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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
−8.549979548628665626271244863535, −7.81859886098920314523684637282, −7.10291963505106693688726108431, −6.29903148779010672865544694424, −5.88861912100809943232135642007, −5.00975131264818069186668197726, −3.53515574294133907235941311663, −2.56659140590734921531727251785, −1.40830415982188810444140062018, −0.906404436886582757210966039553,
0.906404436886582757210966039553, 1.40830415982188810444140062018, 2.56659140590734921531727251785, 3.53515574294133907235941311663, 5.00975131264818069186668197726, 5.88861912100809943232135642007, 6.29903148779010672865544694424, 7.10291963505106693688726108431, 7.81859886098920314523684637282, 8.549979548628665626271244863535
