| L(s) = 1 | + 2.56·2-s + 3-s + 4.60·4-s + 2.15·5-s + 2.56·6-s + 2.66·7-s + 6.68·8-s + 9-s + 5.54·10-s − 5.09·11-s + 4.60·12-s + 13-s + 6.84·14-s + 2.15·15-s + 7.97·16-s − 1.89·17-s + 2.56·18-s − 6.55·19-s + 9.92·20-s + 2.66·21-s − 13.0·22-s + 6.71·23-s + 6.68·24-s − 0.346·25-s + 2.56·26-s + 27-s + 12.2·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.577·3-s + 2.30·4-s + 0.964·5-s + 1.04·6-s + 1.00·7-s + 2.36·8-s + 0.333·9-s + 1.75·10-s − 1.53·11-s + 1.32·12-s + 0.277·13-s + 1.82·14-s + 0.556·15-s + 1.99·16-s − 0.460·17-s + 0.605·18-s − 1.50·19-s + 2.21·20-s + 0.581·21-s − 2.79·22-s + 1.39·23-s + 1.36·24-s − 0.0692·25-s + 0.503·26-s + 0.192·27-s + 2.31·28-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
| \(L(1)\) |
\(\approx\) |
\(9.255009834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.255009834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.56T + 2T^{2} \) |
| 5 | \( 1 - 2.15T + 5T^{2} \) |
| 7 | \( 1 - 2.66T + 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 + 6.55T + 19T^{2} \) |
| 23 | \( 1 - 6.71T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + 9.63T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 + 0.936T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 0.985T + 53T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 - 3.95T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 - 6.72T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 9.47T + 89T^{2} \) |
| 97 | \( 1 + 8.00T + 97T^{2} \) |
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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
−8.326732939817339608554743509342, −7.50932340166850801544647390648, −6.74184788092632744438740393239, −5.99807744810342202883183304110, −5.25315292260952797393738083478, −4.75304902080468737716677462898, −4.04405876246476362656284598976, −2.81711803408319705114103815257, −2.42937260114540527129954189295, −1.58831487767365760027270617152,
1.58831487767365760027270617152, 2.42937260114540527129954189295, 2.81711803408319705114103815257, 4.04405876246476362656284598976, 4.75304902080468737716677462898, 5.25315292260952797393738083478, 5.99807744810342202883183304110, 6.74184788092632744438740393239, 7.50932340166850801544647390648, 8.326732939817339608554743509342
