| L(s) = 1 | + 256·2-s + 6.08e3·3-s + 6.55e4·4-s + 1.25e6·5-s + 1.55e6·6-s − 2.24e7·7-s + 1.67e7·8-s − 9.21e7·9-s + 3.21e8·10-s + 1.72e8·11-s + 3.98e8·12-s − 2.18e9·13-s − 5.75e9·14-s + 7.63e9·15-s + 4.29e9·16-s + 3.01e10·17-s − 2.35e10·18-s − 7.62e10·19-s + 8.22e10·20-s − 1.36e11·21-s + 4.41e10·22-s + 1.30e11·23-s + 1.02e11·24-s + 8.12e11·25-s − 5.58e11·26-s − 1.34e12·27-s − 1.47e12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.535·3-s + 1/2·4-s + 1.43·5-s + 0.378·6-s − 1.47·7-s + 0.353·8-s − 0.713·9-s + 1.01·10-s + 0.242·11-s + 0.267·12-s − 0.741·13-s − 1.04·14-s + 0.769·15-s + 1/4·16-s + 1.04·17-s − 0.504·18-s − 1.03·19-s + 0.718·20-s − 0.788·21-s + 0.171·22-s + 0.347·23-s + 0.189·24-s + 1.06·25-s − 0.524·26-s − 0.917·27-s − 0.736·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.453091468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453091468\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{8} T \) |
| good | 3 | \( 1 - 676 p^{2} T + p^{17} T^{2} \) |
| 5 | \( 1 - 251022 p T + p^{17} T^{2} \) |
| 7 | \( 1 + 458488 p^{2} T + p^{17} T^{2} \) |
| 11 | \( 1 - 172399692 T + p^{17} T^{2} \) |
| 13 | \( 1 + 167703802 p T + p^{17} T^{2} \) |
| 17 | \( 1 - 30163933458 T + p^{17} T^{2} \) |
| 19 | \( 1 + 76275766060 T + p^{17} T^{2} \) |
| 23 | \( 1 - 130466597784 T + p^{17} T^{2} \) |
| 29 | \( 1 - 27694291830 p T + p^{17} T^{2} \) |
| 31 | \( 1 - 2045336056352 T + p^{17} T^{2} \) |
| 37 | \( 1 - 33855367078118 T + p^{17} T^{2} \) |
| 41 | \( 1 - 53206442755242 T + p^{17} T^{2} \) |
| 43 | \( 1 - 26590357792364 T + p^{17} T^{2} \) |
| 47 | \( 1 + 232565394320592 T + p^{17} T^{2} \) |
| 53 | \( 1 + 163277861935626 T + p^{17} T^{2} \) |
| 59 | \( 1 - 697820734313340 T + p^{17} T^{2} \) |
| 61 | \( 1 + 898968337037698 T + p^{17} T^{2} \) |
| 67 | \( 1 + 2667002109080572 T + p^{17} T^{2} \) |
| 71 | \( 1 - 3910637666678472 T + p^{17} T^{2} \) |
| 73 | \( 1 - 5855931724867274 T + p^{17} T^{2} \) |
| 79 | \( 1 + 23821740190145200 T + p^{17} T^{2} \) |
| 83 | \( 1 + 13915745478008556 T + p^{17} T^{2} \) |
| 89 | \( 1 + 30722744829110310 T + p^{17} T^{2} \) |
| 97 | \( 1 - 57649100896826978 T + p^{17} T^{2} \) |
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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
−25.16367047386747969133022572849, −22.77650057109693261278142379133, −21.34585297942020791466669001938, −19.53256631540548305882124070340, −16.87391271811016277101273162591, −14.42047229473740982910507152221, −12.95042648552640109912683385534, −9.700931521273143182455695779559, −6.09606962482516881358734193045, −2.73797173310654543301152857738,
2.73797173310654543301152857738, 6.09606962482516881358734193045, 9.700931521273143182455695779559, 12.95042648552640109912683385534, 14.42047229473740982910507152221, 16.87391271811016277101273162591, 19.53256631540548305882124070340, 21.34585297942020791466669001938, 22.77650057109693261278142379133, 25.16367047386747969133022572849
