| L(s) = 1 | + 8·2-s + 66·3-s + 64·4-s + 400·5-s + 528·6-s + 512·8-s + 2.16e3·9-s + 3.20e3·10-s + 40·11-s + 4.22e3·12-s + 4.45e3·13-s + 2.64e4·15-s + 4.09e3·16-s − 3.65e4·17-s + 1.73e4·18-s + 4.62e4·19-s + 2.56e4·20-s + 320·22-s − 1.05e5·23-s + 3.37e4·24-s + 8.18e4·25-s + 3.56e4·26-s − 1.18e3·27-s − 1.26e5·29-s + 2.11e5·30-s + 1.70e5·31-s + 3.27e4·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.41·3-s + 1/2·4-s + 1.43·5-s + 0.997·6-s + 0.353·8-s + 0.991·9-s + 1.01·10-s + 0.00906·11-s + 0.705·12-s + 0.562·13-s + 2.01·15-s + 1/4·16-s − 1.80·17-s + 0.701·18-s + 1.54·19-s + 0.715·20-s + 0.00640·22-s − 1.80·23-s + 0.498·24-s + 1.04·25-s + 0.397·26-s − 0.0116·27-s − 0.961·29-s + 1.42·30-s + 1.03·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.222582619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.222582619\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 22 p T + p^{7} T^{2} \) |
| 5 | \( 1 - 16 p^{2} T + p^{7} T^{2} \) |
| 11 | \( 1 - 40 T + p^{7} T^{2} \) |
| 13 | \( 1 - 4452 T + p^{7} T^{2} \) |
| 17 | \( 1 + 36502 T + p^{7} T^{2} \) |
| 19 | \( 1 - 46222 T + p^{7} T^{2} \) |
| 23 | \( 1 + 105200 T + p^{7} T^{2} \) |
| 29 | \( 1 + 126334 T + p^{7} T^{2} \) |
| 31 | \( 1 - 170964 T + p^{7} T^{2} \) |
| 37 | \( 1 - 20954 T + p^{7} T^{2} \) |
| 41 | \( 1 + 318486 T + p^{7} T^{2} \) |
| 43 | \( 1 - 1808 p T + p^{7} T^{2} \) |
| 47 | \( 1 + 703716 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1603278 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1171894 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2068872 T + p^{7} T^{2} \) |
| 67 | \( 1 + 994268 T + p^{7} T^{2} \) |
| 71 | \( 1 - 33280 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2971454 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2376168 T + p^{7} T^{2} \) |
| 83 | \( 1 - 2122358 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6920346 T + p^{7} T^{2} \) |
| 97 | \( 1 + 4952710 T + p^{7} T^{2} \) |
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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
−13.19888305996452881642293430251, −11.59127910014426213274675374016, −10.11294664617784647525057815706, −9.284397287774926212166659615472, −8.185293042809333906115854677888, −6.72295146865447405457477080806, −5.54703453094958019703171413798, −3.94097565906488814187089291419, −2.59169764339749171510036926564, −1.75090341581876005814136355640,
1.75090341581876005814136355640, 2.59169764339749171510036926564, 3.94097565906488814187089291419, 5.54703453094958019703171413798, 6.72295146865447405457477080806, 8.185293042809333906115854677888, 9.284397287774926212166659615472, 10.11294664617784647525057815706, 11.59127910014426213274675374016, 13.19888305996452881642293430251
