| L(s) = 1 | + 9.59·2-s + 6.30·3-s + 60.0·4-s + 60.5·6-s − 230.·7-s + 268.·8-s − 203.·9-s − 360.·11-s + 378.·12-s + 331.·13-s − 2.20e3·14-s + 657.·16-s + 1.93e3·17-s − 1.94e3·18-s + 2.66e3·19-s − 1.45e3·21-s − 3.45e3·22-s − 442.·23-s + 1.69e3·24-s + 3.18e3·26-s − 2.81e3·27-s − 1.38e4·28-s + 367.·29-s + 8.81e3·31-s − 2.29e3·32-s − 2.27e3·33-s + 1.85e4·34-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 0.404·3-s + 1.87·4-s + 0.686·6-s − 1.77·7-s + 1.48·8-s − 0.836·9-s − 0.898·11-s + 0.758·12-s + 0.544·13-s − 3.01·14-s + 0.641·16-s + 1.62·17-s − 1.41·18-s + 1.69·19-s − 0.718·21-s − 1.52·22-s − 0.174·23-s + 0.600·24-s + 0.923·26-s − 0.743·27-s − 3.32·28-s + 0.0811·29-s + 1.64·31-s − 0.396·32-s − 0.363·33-s + 2.75·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.911400081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.911400081\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 - 9.59T + 32T^{2} \) |
| 3 | \( 1 - 6.30T + 243T^{2} \) |
| 7 | \( 1 + 230.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 360.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 331.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.93e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 442.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 367.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.24e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 430.T + 1.15e8T^{2} \) |
| 47 | \( 1 - 1.67e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.76e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.88e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.91e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.51e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.98e4T + 8.58e9T^{2} \) |
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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
−9.372738159968226061587908555983, −8.048312069086074341955932352271, −7.29627375826143292809878393584, −6.06192857852759450070412462427, −5.90944959169229128994489962893, −4.88558276040047484081933481979, −3.55592436851057414761740413437, −3.14503240775175392183954747852, −2.59293851241136568864799627051, −0.76351679346929566152373073240,
0.76351679346929566152373073240, 2.59293851241136568864799627051, 3.14503240775175392183954747852, 3.55592436851057414761740413437, 4.88558276040047484081933481979, 5.90944959169229128994489962893, 6.06192857852759450070412462427, 7.29627375826143292809878393584, 8.048312069086074341955932352271, 9.372738159968226061587908555983
