| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 146.·5-s + 216·6-s + 343·7-s + 512·8-s + 729·9-s − 1.17e3·10-s − 860.·11-s + 1.72e3·12-s + 2.19e3·13-s + 2.74e3·14-s − 3.96e3·15-s + 4.09e3·16-s − 6.99e3·17-s + 5.83e3·18-s + 1.29e3·19-s − 9.40e3·20-s + 9.26e3·21-s − 6.88e3·22-s + 6.45e4·23-s + 1.38e4·24-s − 5.65e4·25-s + 1.75e4·26-s + 1.96e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.525·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.371·10-s − 0.194·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.303·15-s + 0.250·16-s − 0.345·17-s + 0.235·18-s + 0.0432·19-s − 0.262·20-s + 0.218·21-s − 0.137·22-s + 1.10·23-s + 0.204·24-s − 0.723·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.647800016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.647800016\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 5 | \( 1 + 146.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 860.T + 1.94e7T^{2} \) |
| 17 | \( 1 + 6.99e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.29e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.45e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.35e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.44e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.93e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.43e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.17e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.04e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.17e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 7.93e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.33e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.55e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.60e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.30e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.17e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.14e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.25e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.47e7T + 8.07e13T^{2} \) |
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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
−9.641775660678360047632576192060, −8.632965358422377460937254698546, −7.81744776128553164667315437949, −7.05007668084238921049795695025, −5.94869911440921175286555740882, −4.83045282301163144375671199099, −4.03029932665594061123072980226, −3.07154555821164861528706105985, −2.07256784211713192685944876944, −0.819982658127725961918453150177,
0.819982658127725961918453150177, 2.07256784211713192685944876944, 3.07154555821164861528706105985, 4.03029932665594061123072980226, 4.83045282301163144375671199099, 5.94869911440921175286555740882, 7.05007668084238921049795695025, 7.81744776128553164667315437949, 8.632965358422377460937254698546, 9.641775660678360047632576192060
