L(s) = 1 | + 2.11·2-s + 3-s + 2.49·4-s − 0.696·5-s + 2.11·6-s + 5.08·7-s + 1.04·8-s + 9-s − 1.47·10-s + 2.93·11-s + 2.49·12-s + 13-s + 10.7·14-s − 0.696·15-s − 2.77·16-s + 6.82·17-s + 2.11·18-s + 3.34·19-s − 1.73·20-s + 5.08·21-s + 6.22·22-s − 5.43·23-s + 1.04·24-s − 4.51·25-s + 2.11·26-s + 27-s + 12.6·28-s + ⋯ |
L(s) = 1 | + 1.49·2-s + 0.577·3-s + 1.24·4-s − 0.311·5-s + 0.865·6-s + 1.92·7-s + 0.367·8-s + 0.333·9-s − 0.466·10-s + 0.885·11-s + 0.719·12-s + 0.277·13-s + 2.88·14-s − 0.179·15-s − 0.694·16-s + 1.65·17-s + 0.499·18-s + 0.767·19-s − 0.387·20-s + 1.11·21-s + 1.32·22-s − 1.13·23-s + 0.212·24-s − 0.903·25-s + 0.415·26-s + 0.192·27-s + 2.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.907887850\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.907887850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 5 | \( 1 + 0.696T + 5T^{2} \) |
| 7 | \( 1 - 5.08T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 + 5.43T + 23T^{2} \) |
| 29 | \( 1 + 7.09T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 + 9.02T + 37T^{2} \) |
| 41 | \( 1 - 0.0904T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 6.36T + 47T^{2} \) |
| 53 | \( 1 - 0.641T + 53T^{2} \) |
| 59 | \( 1 + 5.15T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 - 1.75T + 67T^{2} \) |
| 71 | \( 1 + 1.72T + 71T^{2} \) |
| 73 | \( 1 - 0.254T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + 5.88T + 83T^{2} \) |
| 89 | \( 1 - 7.00T + 89T^{2} \) |
| 97 | \( 1 + 4.49T + 97T^{2} \) |
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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
−8.160998743675995796767309043541, −7.74394922414132318442788350959, −6.97785267972746937635451347324, −5.85974961575837531013736077086, −5.34369958010305415310331044579, −4.60465075325704376390596184826, −3.80496280662821409687385163639, −3.42522578356670443267851900841, −2.08421641985953430952903691976, −1.40420478709113862439259181702,
1.40420478709113862439259181702, 2.08421641985953430952903691976, 3.42522578356670443267851900841, 3.80496280662821409687385163639, 4.60465075325704376390596184826, 5.34369958010305415310331044579, 5.85974961575837531013736077086, 6.97785267972746937635451347324, 7.74394922414132318442788350959, 8.160998743675995796767309043541