| L(s) = 1 | − 27.7·3-s − 25·5-s − 49·7-s + 526.·9-s + 15.5·11-s − 603.·13-s + 693.·15-s − 2.24e3·17-s − 3.09e3·19-s + 1.35e3·21-s − 3.53e3·23-s + 625·25-s − 7.87e3·27-s + 2.01e3·29-s − 2.92e3·31-s − 431.·33-s + 1.22e3·35-s + 1.19e4·37-s + 1.67e4·39-s + 2.90e3·41-s − 1.92e4·43-s − 1.31e4·45-s + 459.·47-s + 2.40e3·49-s + 6.22e4·51-s − 2.65e4·53-s − 388.·55-s + ⋯ |
| L(s) = 1 | − 1.77·3-s − 0.447·5-s − 0.377·7-s + 2.16·9-s + 0.0387·11-s − 0.990·13-s + 0.795·15-s − 1.88·17-s − 1.96·19-s + 0.672·21-s − 1.39·23-s + 0.200·25-s − 2.07·27-s + 0.444·29-s − 0.546·31-s − 0.0689·33-s + 0.169·35-s + 1.43·37-s + 1.76·39-s + 0.269·41-s − 1.58·43-s − 0.969·45-s + 0.0303·47-s + 0.142·49-s + 3.34·51-s − 1.29·53-s − 0.0173·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1091718641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1091718641\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 27.7T + 243T^{2} \) |
| 11 | \( 1 - 15.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 603.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.24e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 3.09e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.19e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.90e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 459.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.57e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.18e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.44e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.08e4T + 8.58e9T^{2} \) |
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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
−11.01116111465784731596219407835, −10.43977423751863804017047431821, −9.326650058547690269257352166441, −7.917310093086639751995424153110, −6.61649795245467622708314681370, −6.27220707141396356483205874696, −4.82106404147341616184262581478, −4.19958753551216650697715036816, −2.07956137909602239816666221932, −0.19520177569918713715354960241,
0.19520177569918713715354960241, 2.07956137909602239816666221932, 4.19958753551216650697715036816, 4.82106404147341616184262581478, 6.27220707141396356483205874696, 6.61649795245467622708314681370, 7.917310093086639751995424153110, 9.326650058547690269257352166441, 10.43977423751863804017047431821, 11.01116111465784731596219407835
