| L(s) = 1 | + 6.97·2-s + 81·3-s − 463.·4-s + 1.65e3·5-s + 564.·6-s + 2.03e3·7-s − 6.80e3·8-s + 6.56e3·9-s + 1.15e4·10-s + 9.31e3·11-s − 3.75e4·12-s − 1.11e5·13-s + 1.42e4·14-s + 1.34e5·15-s + 1.89e5·16-s − 4.99e5·17-s + 4.57e4·18-s + 5.72e5·19-s − 7.67e5·20-s + 1.65e5·21-s + 6.49e4·22-s − 1.77e6·23-s − 5.50e5·24-s + 7.90e5·25-s − 7.76e5·26-s + 5.31e5·27-s − 9.44e5·28-s + ⋯ |
| L(s) = 1 | + 0.308·2-s + 0.577·3-s − 0.905·4-s + 1.18·5-s + 0.177·6-s + 0.320·7-s − 0.587·8-s + 0.333·9-s + 0.365·10-s + 0.191·11-s − 0.522·12-s − 1.08·13-s + 0.0988·14-s + 0.684·15-s + 0.724·16-s − 1.45·17-s + 0.102·18-s + 1.00·19-s − 1.07·20-s + 0.185·21-s + 0.0591·22-s − 1.32·23-s − 0.338·24-s + 0.404·25-s − 0.333·26-s + 0.192·27-s − 0.290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 6.97T + 512T^{2} \) |
| 5 | \( 1 - 1.65e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.03e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 9.31e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.11e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.99e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.72e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.77e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.11e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.68e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.12e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.80e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.86e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 8.20e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.44e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 9.11e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.68e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.59e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.80e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.32e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.86e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.77e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.52e8T + 7.60e17T^{2} \) |
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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
−10.10275840646508159916319864442, −9.524190816963143521481805008259, −8.718254231544706547342727568517, −7.51416419393818714873599763383, −6.11489775633363560446459830987, −5.08786749996210033668551113546, −4.11335264778647375633903361631, −2.66828251076541522505431583254, −1.62925308719520663546726775519, 0,
1.62925308719520663546726775519, 2.66828251076541522505431583254, 4.11335264778647375633903361631, 5.08786749996210033668551113546, 6.11489775633363560446459830987, 7.51416419393818714873599763383, 8.718254231544706547342727568517, 9.524190816963143521481805008259, 10.10275840646508159916319864442
