| L(s) = 1 | + 3-s − 3.88·5-s − 7-s + 9-s − 3.21·11-s + 5.39·13-s − 3.88·15-s − 4.41·17-s − 19-s − 21-s + 3.39·23-s + 10.1·25-s + 27-s − 7.28·29-s − 0.187·31-s − 3.21·33-s + 3.88·35-s − 4.60·37-s + 5.39·39-s − 9.77·41-s − 3.88·45-s − 4.45·47-s + 49-s − 4.41·51-s + 11.2·53-s + 12.4·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.73·5-s − 0.377·7-s + 0.333·9-s − 0.968·11-s + 1.49·13-s − 1.00·15-s − 1.06·17-s − 0.229·19-s − 0.218·21-s + 0.708·23-s + 2.02·25-s + 0.192·27-s − 1.35·29-s − 0.0336·31-s − 0.558·33-s + 0.657·35-s − 0.757·37-s + 0.864·39-s − 1.52·41-s − 0.579·45-s − 0.649·47-s + 0.142·49-s − 0.617·51-s + 1.55·53-s + 1.68·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.082877918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082877918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.88T + 5T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 - 5.39T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 + 7.28T + 29T^{2} \) |
| 31 | \( 1 + 0.187T + 31T^{2} \) |
| 37 | \( 1 + 4.60T + 37T^{2} \) |
| 41 | \( 1 + 9.77T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4.45T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 5.77T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.08T + 73T^{2} \) |
| 79 | \( 1 + 2.68T + 79T^{2} \) |
| 83 | \( 1 + 3.92T + 83T^{2} \) |
| 89 | \( 1 - 2.70T + 89T^{2} \) |
| 97 | \( 1 + 0.227T + 97T^{2} \) |
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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
−8.180147352473643151421308468050, −7.24861754501999214291783797990, −7.00114881667931914955245309629, −5.95438544259550959605505953437, −4.98297152109937953216006489351, −4.19610252724115562262987292962, −3.56281979919637936154614597714, −3.07188853223184423391886889393, −1.90036113078400727492739229611, −0.50909016503966996239391844152,
0.50909016503966996239391844152, 1.90036113078400727492739229611, 3.07188853223184423391886889393, 3.56281979919637936154614597714, 4.19610252724115562262987292962, 4.98297152109937953216006489351, 5.95438544259550959605505953437, 7.00114881667931914955245309629, 7.24861754501999214291783797990, 8.180147352473643151421308468050
