L(s) = 1 | − 2.16·2-s − 3-s + 2.70·4-s + 0.657·5-s + 2.16·6-s − 0.817·7-s − 1.52·8-s + 9-s − 1.42·10-s + 0.238·11-s − 2.70·12-s − 13-s + 1.77·14-s − 0.657·15-s − 2.09·16-s − 7.29·17-s − 2.16·18-s − 7.89·19-s + 1.77·20-s + 0.817·21-s − 0.516·22-s − 4.80·23-s + 1.52·24-s − 4.56·25-s + 2.16·26-s − 27-s − 2.21·28-s + ⋯ |
L(s) = 1 | − 1.53·2-s − 0.577·3-s + 1.35·4-s + 0.294·5-s + 0.885·6-s − 0.309·7-s − 0.540·8-s + 0.333·9-s − 0.451·10-s + 0.0718·11-s − 0.780·12-s − 0.277·13-s + 0.474·14-s − 0.169·15-s − 0.523·16-s − 1.76·17-s − 0.511·18-s − 1.81·19-s + 0.397·20-s + 0.178·21-s − 0.110·22-s − 1.00·23-s + 0.312·24-s − 0.913·25-s + 0.425·26-s − 0.192·27-s − 0.418·28-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
\(L(1)\) |
\(\approx\) |
\(0.2366220848\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2366220848\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.16T + 2T^{2} \) |
| 5 | \( 1 - 0.657T + 5T^{2} \) |
| 7 | \( 1 + 0.817T + 7T^{2} \) |
| 11 | \( 1 - 0.238T + 11T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 + 7.89T + 19T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 + 5.74T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 8.40T + 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 1.14T + 47T^{2} \) |
| 53 | \( 1 - 7.78T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 + 0.371T + 71T^{2} \) |
| 73 | \( 1 - 8.08T + 73T^{2} \) |
| 79 | \( 1 - 8.84T + 79T^{2} \) |
| 83 | \( 1 + 6.90T + 83T^{2} \) |
| 89 | \( 1 - 2.55T + 89T^{2} \) |
| 97 | \( 1 + 5.02T + 97T^{2} \) |
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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
−8.555665880416234618597947850923, −7.86673963769102771877719525774, −7.03428377729234629078565774704, −6.43241321352862824662570348001, −5.86817642630858594790644855046, −4.61154163449604862523441392293, −3.96590240611571072967260485399, −2.27963434659847630537943947368, −1.89951106922500970910685791002, −0.34343697764120037074694862512,
0.34343697764120037074694862512, 1.89951106922500970910685791002, 2.27963434659847630537943947368, 3.96590240611571072967260485399, 4.61154163449604862523441392293, 5.86817642630858594790644855046, 6.43241321352862824662570348001, 7.03428377729234629078565774704, 7.86673963769102771877719525774, 8.555665880416234618597947850923