| L(s) = 1 | − 5·3-s + 9·5-s − 2·9-s + 57·11-s + 70·13-s − 45·15-s − 51·17-s + 5·19-s − 69·23-s − 44·25-s + 145·27-s + 114·29-s + 23·31-s − 285·33-s − 253·37-s − 350·39-s + 42·41-s + 124·43-s − 18·45-s + 201·47-s + 255·51-s − 393·53-s + 513·55-s − 25·57-s + 219·59-s + 709·61-s + 630·65-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 0.804·5-s − 0.0740·9-s + 1.56·11-s + 1.49·13-s − 0.774·15-s − 0.727·17-s + 0.0603·19-s − 0.625·23-s − 0.351·25-s + 1.03·27-s + 0.729·29-s + 0.133·31-s − 1.50·33-s − 1.12·37-s − 1.43·39-s + 0.159·41-s + 0.439·43-s − 0.0596·45-s + 0.623·47-s + 0.700·51-s − 1.01·53-s + 1.25·55-s − 0.0580·57-s + 0.483·59-s + 1.48·61-s + 1.20·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.835986524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835986524\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 5 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 114 T + p^{3} T^{2} \) |
| 31 | \( 1 - 23 T + p^{3} T^{2} \) |
| 37 | \( 1 + 253 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 124 T + p^{3} T^{2} \) |
| 47 | \( 1 - 201 T + p^{3} T^{2} \) |
| 53 | \( 1 + 393 T + p^{3} T^{2} \) |
| 59 | \( 1 - 219 T + p^{3} T^{2} \) |
| 61 | \( 1 - 709 T + p^{3} T^{2} \) |
| 67 | \( 1 + 419 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 313 T + p^{3} T^{2} \) |
| 79 | \( 1 + 461 T + p^{3} T^{2} \) |
| 83 | \( 1 + 588 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1017 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1834 T + p^{3} T^{2} \) |
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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
−10.00068401813598547236388508855, −9.028521445973491576918222474442, −8.452990307718546266201831828538, −6.90883437607007374537154629551, −6.19509565931142792222231728836, −5.79685773828123009783716870688, −4.54878839767810093629697884582, −3.52964449604414804826757508467, −1.90832021112420557370008271482, −0.835841188498416700760494647301,
0.835841188498416700760494647301, 1.90832021112420557370008271482, 3.52964449604414804826757508467, 4.54878839767810093629697884582, 5.79685773828123009783716870688, 6.19509565931142792222231728836, 6.90883437607007374537154629551, 8.452990307718546266201831828538, 9.028521445973491576918222474442, 10.00068401813598547236388508855
