| L(s) = 1 | − 252·3-s − 4.83e3·5-s − 1.67e4·7-s − 1.13e5·9-s − 5.34e5·11-s + 5.77e5·13-s + 1.21e6·15-s − 6.90e6·17-s − 1.06e7·19-s + 4.21e6·21-s + 1.86e7·23-s − 2.54e7·25-s + 7.32e7·27-s − 1.28e8·29-s − 5.28e7·31-s + 1.34e8·33-s + 8.08e7·35-s + 1.82e8·37-s − 1.45e8·39-s + 3.08e8·41-s + 1.71e7·43-s + 5.48e8·45-s + 2.68e9·47-s − 1.69e9·49-s + 1.74e9·51-s + 1.59e9·53-s + 2.58e9·55-s + ⋯ |
| L(s) = 1 | − 0.598·3-s − 0.691·5-s − 0.376·7-s − 0.641·9-s − 1.00·11-s + 0.431·13-s + 0.413·15-s − 1.17·17-s − 0.987·19-s + 0.225·21-s + 0.603·23-s − 0.522·25-s + 0.982·27-s − 1.16·29-s − 0.331·31-s + 0.599·33-s + 0.260·35-s + 0.431·37-s − 0.258·39-s + 0.415·41-s + 0.0177·43-s + 0.443·45-s + 1.70·47-s − 0.858·49-s + 0.706·51-s + 0.524·53-s + 0.691·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.4922889527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4922889527\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 28 p^{2} T + p^{11} T^{2} \) |
| 5 | \( 1 + 966 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 2392 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 534612 T + p^{11} T^{2} \) |
| 13 | \( 1 - 577738 T + p^{11} T^{2} \) |
| 17 | \( 1 + 6905934 T + p^{11} T^{2} \) |
| 19 | \( 1 + 10661420 T + p^{11} T^{2} \) |
| 23 | \( 1 - 18643272 T + p^{11} T^{2} \) |
| 29 | \( 1 + 128406630 T + p^{11} T^{2} \) |
| 31 | \( 1 + 52843168 T + p^{11} T^{2} \) |
| 37 | \( 1 - 182213314 T + p^{11} T^{2} \) |
| 41 | \( 1 - 308120442 T + p^{11} T^{2} \) |
| 43 | \( 1 - 17125708 T + p^{11} T^{2} \) |
| 47 | \( 1 - 2687348496 T + p^{11} T^{2} \) |
| 53 | \( 1 - 1596055698 T + p^{11} T^{2} \) |
| 59 | \( 1 - 5189203740 T + p^{11} T^{2} \) |
| 61 | \( 1 + 6956478662 T + p^{11} T^{2} \) |
| 67 | \( 1 - 15481826884 T + p^{11} T^{2} \) |
| 71 | \( 1 - 9791485272 T + p^{11} T^{2} \) |
| 73 | \( 1 - 1463791322 T + p^{11} T^{2} \) |
| 79 | \( 1 - 38116845680 T + p^{11} T^{2} \) |
| 83 | \( 1 - 29335099668 T + p^{11} T^{2} \) |
| 89 | \( 1 + 24992917110 T + p^{11} T^{2} \) |
| 97 | \( 1 - 75013568546 T + p^{11} T^{2} \) |
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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
−12.56503441437447836636434735323, −11.29775636139093780999153693005, −10.71821457280364394924338738682, −9.060871021926277876623605490261, −7.918418975024888782755911718513, −6.54824631052139305300310908857, −5.36613183261381513224257199660, −3.95971436108400539432925650934, −2.44637632417030839942559200276, −0.38335448236615559328357671822,
0.38335448236615559328357671822, 2.44637632417030839942559200276, 3.95971436108400539432925650934, 5.36613183261381513224257199660, 6.54824631052139305300310908857, 7.918418975024888782755911718513, 9.060871021926277876623605490261, 10.71821457280364394924338738682, 11.29775636139093780999153693005, 12.56503441437447836636434735323
