| L(s) = 1 | + 5.65·2-s − 15.5·3-s + 32.0·4-s + 38.5i·5-s − 88.1·6-s + 40.4i·7-s + 181.·8-s + 243·9-s + 217. i·10-s − 1.76e3i·11-s − 498.·12-s + 152.·13-s + 228. i·14-s − 600. i·15-s + 1.02e3·16-s + 1.22e3i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.500·4-s + 0.308i·5-s − 0.408·6-s + 0.117i·7-s + 0.353·8-s + 0.333·9-s + 0.217i·10-s − 1.32i·11-s − 0.288·12-s + 0.0695·13-s + 0.0833i·14-s − 0.177i·15-s + 0.250·16-s + 0.249i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0591i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.998 - 0.0591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.685984133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.685984133\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65T \) |
| 3 | \( 1 + 15.5T \) |
| 23 | \( 1 + (-1.21e4 + 720. i)T \) |
| good | 5 | \( 1 - 38.5iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 40.4iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.76e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 152.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 1.22e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 9.28e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 - 7.70e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.59e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 8.95e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.08e5T + 4.75e9T^{2} \) |
| 43 | \( 1 - 7.11e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.01e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 5.08e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 7.86e4T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.43e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.69e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.89e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 1.03e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 8.82e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 3.86e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.13e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.84e4iT - 8.32e11T^{2} \) |
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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
−12.12179023030498855852719722852, −11.07621347792995760318120941514, −10.46531738690592618231336816589, −8.885578777065042354571237271492, −7.56321993358859272311792040493, −6.27246936129068391439914888263, −5.56245933061335895904515234213, −4.11041606843321131798107806185, −2.83421771475356808163319569018, −0.983825770719667982056706346960,
0.986851169133387407994662542659, 2.65442889391439304966117502239, 4.43936297442184047155603213281, 5.09249545565437461351952212214, 6.57358963752598612160385562918, 7.37286498507248648011318045341, 8.998644538830150382163850051498, 10.20195522855814139550558263643, 11.21581577970772003290449822550, 12.18892419687275347196001762845
