| L(s) = 1 | + 2-s − 3·3-s − 7·4-s − 3·6-s + 20·7-s − 15·8-s + 9·9-s − 4·11-s + 21·12-s + 76·13-s + 20·14-s + 41·16-s − 22·17-s + 9·18-s − 19·19-s − 60·21-s − 4·22-s − 82·23-s + 45·24-s + 76·26-s − 27·27-s − 140·28-s + 242·29-s − 126·31-s + 161·32-s + 12·33-s − 22·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.204·6-s + 1.07·7-s − 0.662·8-s + 1/3·9-s − 0.109·11-s + 0.505·12-s + 1.62·13-s + 0.381·14-s + 0.640·16-s − 0.313·17-s + 0.117·18-s − 0.229·19-s − 0.623·21-s − 0.0387·22-s − 0.743·23-s + 0.382·24-s + 0.573·26-s − 0.192·27-s − 0.944·28-s + 1.54·29-s − 0.730·31-s + 0.889·32-s + 0.0633·33-s − 0.110·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.837327359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837327359\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 76 T + p^{3} T^{2} \) |
| 17 | \( 1 + 22 T + p^{3} T^{2} \) |
| 23 | \( 1 + 82 T + p^{3} T^{2} \) |
| 29 | \( 1 - 242 T + p^{3} T^{2} \) |
| 31 | \( 1 + 126 T + p^{3} T^{2} \) |
| 37 | \( 1 - 180 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 308 T + p^{3} T^{2} \) |
| 47 | \( 1 - 522 T + p^{3} T^{2} \) |
| 53 | \( 1 - 70 T + p^{3} T^{2} \) |
| 59 | \( 1 - 188 T + p^{3} T^{2} \) |
| 61 | \( 1 + 706 T + p^{3} T^{2} \) |
| 67 | \( 1 + 104 T + p^{3} T^{2} \) |
| 71 | \( 1 + 432 T + p^{3} T^{2} \) |
| 73 | \( 1 + 718 T + p^{3} T^{2} \) |
| 79 | \( 1 - 94 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1296 T + p^{3} T^{2} \) |
| 89 | \( 1 - 846 T + p^{3} T^{2} \) |
| 97 | \( 1 + 830 T + p^{3} T^{2} \) |
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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
−8.936037909290993793937021656043, −8.492369329725045633135801255828, −7.70067429490741323109597826874, −6.42049643287073054916875546776, −5.80040817651607681198524411644, −4.88016702385162651932414724574, −4.29321963271847227179544870307, −3.35242574681224847912887151219, −1.74186867656097887371559789802, −0.69097678946666844912435471477,
0.69097678946666844912435471477, 1.74186867656097887371559789802, 3.35242574681224847912887151219, 4.29321963271847227179544870307, 4.88016702385162651932414724574, 5.80040817651607681198524411644, 6.42049643287073054916875546776, 7.70067429490741323109597826874, 8.492369329725045633135801255828, 8.936037909290993793937021656043
