| L(s) = 1 | + 2·2-s − 0.113·3-s + 4·4-s − 7.70·5-s − 0.227·6-s − 7·7-s + 8·8-s − 26.9·9-s − 15.4·10-s − 6.76·11-s − 0.455·12-s − 14·14-s + 0.877·15-s + 16·16-s − 5.61·17-s − 53.9·18-s − 68.8·19-s − 30.8·20-s + 0.797·21-s − 13.5·22-s + 178.·23-s − 0.911·24-s − 65.7·25-s + 6.15·27-s − 28·28-s − 189.·29-s + 1.75·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0219·3-s + 0.5·4-s − 0.688·5-s − 0.0155·6-s − 0.377·7-s + 0.353·8-s − 0.999·9-s − 0.487·10-s − 0.185·11-s − 0.0109·12-s − 0.267·14-s + 0.0150·15-s + 0.250·16-s − 0.0801·17-s − 0.706·18-s − 0.830·19-s − 0.344·20-s + 0.00828·21-s − 0.131·22-s + 1.61·23-s − 0.00775·24-s − 0.525·25-s + 0.0438·27-s − 0.188·28-s − 1.21·29-s + 0.0106·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.802482292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802482292\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.113T + 27T^{2} \) |
| 5 | \( 1 + 7.70T + 125T^{2} \) |
| 11 | \( 1 + 6.76T + 1.33e3T^{2} \) |
| 17 | \( 1 + 5.61T + 4.91e3T^{2} \) |
| 19 | \( 1 + 68.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 502.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 18.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 685.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 159.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 661.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 617.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 65.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 825.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 298.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 461.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 450.T + 9.12e5T^{2} \) |
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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
−8.781961734412262513671745928368, −7.54829367972243810462577619421, −7.22717744775498182485797140526, −6.04457185025514734578337237037, −5.58582046001389437656493826966, −4.56899575368477643801720855967, −3.73964253417884817261290444393, −3.01344924692726405818907743491, −2.06533869065877337775361642269, −0.50873470615502136646058164797,
0.50873470615502136646058164797, 2.06533869065877337775361642269, 3.01344924692726405818907743491, 3.73964253417884817261290444393, 4.56899575368477643801720855967, 5.58582046001389437656493826966, 6.04457185025514734578337237037, 7.22717744775498182485797140526, 7.54829367972243810462577619421, 8.781961734412262513671745928368
