| L(s) = 1 | − 2.24·2-s − 1.80·3-s + 3.02·4-s + 4.04·6-s + 5.11·7-s − 2.30·8-s + 0.248·9-s + 4.18·11-s − 5.45·12-s + 5.65·13-s − 11.4·14-s − 0.884·16-s − 0.983·17-s − 0.556·18-s + 0.188·19-s − 9.22·21-s − 9.38·22-s + 7.24·23-s + 4.15·24-s − 12.6·26-s + 4.95·27-s + 15.5·28-s + 4.85·29-s − 6.91·31-s + 6.59·32-s − 7.54·33-s + 2.20·34-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 1.04·3-s + 1.51·4-s + 1.64·6-s + 1.93·7-s − 0.815·8-s + 0.0827·9-s + 1.26·11-s − 1.57·12-s + 1.56·13-s − 3.06·14-s − 0.221·16-s − 0.238·17-s − 0.131·18-s + 0.0432·19-s − 2.01·21-s − 2.00·22-s + 1.51·23-s + 0.848·24-s − 2.48·26-s + 0.954·27-s + 2.92·28-s + 0.901·29-s − 1.24·31-s + 1.16·32-s − 1.31·33-s + 0.378·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.052906490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.052906490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 + 1.80T + 3T^{2} \) |
| 7 | \( 1 - 5.11T + 7T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 0.983T + 17T^{2} \) |
| 19 | \( 1 - 0.188T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + 6.91T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 - 6.75T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + 7.51T + 71T^{2} \) |
| 73 | \( 1 - 3.90T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 - 1.73T + 97T^{2} \) |
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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
−8.441925604158427561383470540527, −7.32602206114182485721765788118, −6.95899006832369547336937826357, −6.08537325644631996957749527258, −5.35601212195686649899543879760, −4.61116346731937300395907890291, −3.70045349383408237254027250668, −2.18214660818792717596152718803, −1.25068257162036416352426928088, −0.915689016781422572948777303681,
0.915689016781422572948777303681, 1.25068257162036416352426928088, 2.18214660818792717596152718803, 3.70045349383408237254027250668, 4.61116346731937300395907890291, 5.35601212195686649899543879760, 6.08537325644631996957749527258, 6.95899006832369547336937826357, 7.32602206114182485721765788118, 8.441925604158427561383470540527
