L(s) = 1 | − 2.20·2-s − 3.01·3-s + 2.85·4-s − 5-s + 6.63·6-s − 7-s − 1.87·8-s + 6.07·9-s + 2.20·10-s − 3.78·11-s − 8.58·12-s + 4.33·13-s + 2.20·14-s + 3.01·15-s − 1.57·16-s + 1.42·17-s − 13.3·18-s − 3.78·19-s − 2.85·20-s + 3.01·21-s + 8.32·22-s − 8.43·23-s + 5.63·24-s + 25-s − 9.54·26-s − 9.25·27-s − 2.85·28-s + ⋯ |
L(s) = 1 | − 1.55·2-s − 1.73·3-s + 1.42·4-s − 0.447·5-s + 2.70·6-s − 0.377·7-s − 0.661·8-s + 2.02·9-s + 0.696·10-s − 1.14·11-s − 2.47·12-s + 1.20·13-s + 0.588·14-s + 0.777·15-s − 0.394·16-s + 0.344·17-s − 3.15·18-s − 0.868·19-s − 0.637·20-s + 0.657·21-s + 1.77·22-s − 1.75·23-s + 1.15·24-s + 0.200·25-s − 1.87·26-s − 1.78·27-s − 0.538·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01085410241\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01085410241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 - 1.42T + 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 23 | \( 1 + 8.43T + 23T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 + 2.71T + 37T^{2} \) |
| 41 | \( 1 + 1.56T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 + 3.10T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 0.748T + 67T^{2} \) |
| 71 | \( 1 - 1.44T + 71T^{2} \) |
| 73 | \( 1 - 5.14T + 73T^{2} \) |
| 79 | \( 1 - 9.81T + 79T^{2} \) |
| 83 | \( 1 - 0.0758T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 8.60T + 97T^{2} \) |
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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
−7.76111631158649126757649725501, −7.34237675344753564885606752680, −6.42704574405561325855941434983, −6.01503137156385575105358183029, −5.32242047804956720753916589258, −4.36158284284198586715637415405, −3.59234132255556167845486711379, −2.14130026172048840025793830380, −1.30700928970359083492222745742, −0.07805139384320198367112503338,
0.07805139384320198367112503338, 1.30700928970359083492222745742, 2.14130026172048840025793830380, 3.59234132255556167845486711379, 4.36158284284198586715637415405, 5.32242047804956720753916589258, 6.01503137156385575105358183029, 6.42704574405561325855941434983, 7.34237675344753564885606752680, 7.76111631158649126757649725501