| L(s) = 1 | − 2.80·3-s + 0.737·5-s − 3.85·7-s + 4.84·9-s − 3.18·11-s + 3.22·13-s − 2.06·15-s + 4.11·17-s + 5.88·19-s + 10.7·21-s − 4.45·25-s − 5.18·27-s + 0.0202·29-s − 10.0·31-s + 8.93·33-s − 2.84·35-s − 2.39·37-s − 9.04·39-s − 8.31·41-s − 7.88·43-s + 3.57·45-s − 1.14·47-s + 7.82·49-s − 11.5·51-s − 1.56·53-s − 2.35·55-s − 16.4·57-s + ⋯ |
| L(s) = 1 | − 1.61·3-s + 0.329·5-s − 1.45·7-s + 1.61·9-s − 0.961·11-s + 0.895·13-s − 0.533·15-s + 0.999·17-s + 1.35·19-s + 2.35·21-s − 0.891·25-s − 0.997·27-s + 0.00375·29-s − 1.81·31-s + 1.55·33-s − 0.480·35-s − 0.394·37-s − 1.44·39-s − 1.29·41-s − 1.20·43-s + 0.533·45-s − 0.167·47-s + 1.11·49-s − 1.61·51-s − 0.214·53-s − 0.317·55-s − 2.18·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5908154476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5908154476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 - 0.737T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 29 | \( 1 - 0.0202T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 + 8.31T + 41T^{2} \) |
| 43 | \( 1 + 7.88T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 - 3.01T + 59T^{2} \) |
| 61 | \( 1 + 2.81T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 3.04T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 4.95T + 89T^{2} \) |
| 97 | \( 1 - 9.93T + 97T^{2} \) |
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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
−8.335751172650475526474538586587, −7.36227624243510612941543382785, −6.80143202727274281215863697410, −6.01341544303529416383475800206, −5.54080043979722685738780827287, −5.08720966134277896872869267645, −3.70907181754807104634156830895, −3.17795906525817127583730629372, −1.66634969277132137860146881979, −0.47448293208521989088624572972,
0.47448293208521989088624572972, 1.66634969277132137860146881979, 3.17795906525817127583730629372, 3.70907181754807104634156830895, 5.08720966134277896872869267645, 5.54080043979722685738780827287, 6.01341544303529416383475800206, 6.80143202727274281215863697410, 7.36227624243510612941543382785, 8.335751172650475526474538586587
