L(s) = 1 | + 2.79·2-s − 3-s + 5.83·4-s − 1.94·5-s − 2.79·6-s + 10.7·8-s + 9-s − 5.45·10-s − 2.68·11-s − 5.83·12-s + 4.19·13-s + 1.94·15-s + 18.4·16-s − 0.632·17-s + 2.79·18-s + 5.55·19-s − 11.3·20-s − 7.52·22-s + 23-s − 10.7·24-s − 1.20·25-s + 11.7·26-s − 27-s + 5.57·29-s + 5.45·30-s − 5.59·31-s + 30.0·32-s + ⋯ |
L(s) = 1 | + 1.97·2-s − 0.577·3-s + 2.91·4-s − 0.870·5-s − 1.14·6-s + 3.79·8-s + 0.333·9-s − 1.72·10-s − 0.810·11-s − 1.68·12-s + 1.16·13-s + 0.502·15-s + 4.60·16-s − 0.153·17-s + 0.659·18-s + 1.27·19-s − 2.54·20-s − 1.60·22-s + 0.208·23-s − 2.19·24-s − 0.241·25-s + 2.30·26-s − 0.192·27-s + 1.03·29-s + 0.995·30-s − 1.00·31-s + 5.30·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.550198657\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.550198657\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 + 0.632T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 - 4.11T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 1.98T + 43T^{2} \) |
| 47 | \( 1 + 0.292T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 + 6.28T + 61T^{2} \) |
| 67 | \( 1 - 3.14T + 67T^{2} \) |
| 71 | \( 1 - 6.00T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 1.81T + 89T^{2} \) |
| 97 | \( 1 + 2.95T + 97T^{2} \) |
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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
−7.996979116748602361505512695981, −7.72803562955776671281632738360, −6.76714568687414919770589726647, −6.19418067624137045630841637865, −5.34973526148071827525724271902, −4.88915681598887839469927738487, −3.92163415094295161143472542549, −3.45097530018005864287317100911, −2.46706805048933552252581574356, −1.15468592851428619226280594731,
1.15468592851428619226280594731, 2.46706805048933552252581574356, 3.45097530018005864287317100911, 3.92163415094295161143472542549, 4.88915681598887839469927738487, 5.34973526148071827525724271902, 6.19418067624137045630841637865, 6.76714568687414919770589726647, 7.72803562955776671281632738360, 7.996979116748602361505512695981