| L(s) = 1 | − 16·2-s + 75·3-s + 256·4-s + 1.97e3·5-s − 1.20e3·6-s + 1.01e4·7-s − 4.09e3·8-s − 1.40e4·9-s − 3.16e4·10-s − 1.88e4·11-s + 1.92e4·12-s − 1.61e5·14-s + 1.48e5·15-s + 6.55e4·16-s − 1.42e5·17-s + 2.24e5·18-s − 8.33e4·19-s + 5.06e5·20-s + 7.58e5·21-s + 3.01e5·22-s − 5.36e5·23-s − 3.07e5·24-s + 1.96e6·25-s − 2.53e6·27-s + 2.58e6·28-s − 2.60e6·29-s − 2.37e6·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.534·3-s + 1/2·4-s + 1.41·5-s − 0.378·6-s + 1.59·7-s − 0.353·8-s − 0.714·9-s − 1.00·10-s − 0.388·11-s + 0.267·12-s − 1.12·14-s + 0.757·15-s + 1/4·16-s − 0.413·17-s + 0.505·18-s − 0.146·19-s + 0.708·20-s + 0.851·21-s + 0.274·22-s − 0.399·23-s − 0.189·24-s + 1.00·25-s − 0.916·27-s + 0.796·28-s − 0.682·29-s − 0.535·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p^{4} T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 25 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 1979 T + p^{9} T^{2} \) |
| 7 | \( 1 - 1445 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 18850 T + p^{9} T^{2} \) |
| 17 | \( 1 + 142403 T + p^{9} T^{2} \) |
| 19 | \( 1 + 83302 T + p^{9} T^{2} \) |
| 23 | \( 1 + 23328 p T + p^{9} T^{2} \) |
| 29 | \( 1 + 2600442 T + p^{9} T^{2} \) |
| 31 | \( 1 - 2214004 T + p^{9} T^{2} \) |
| 37 | \( 1 + 18099241 T + p^{9} T^{2} \) |
| 41 | \( 1 + 26812240 T + p^{9} T^{2} \) |
| 43 | \( 1 + 42253475 T + p^{9} T^{2} \) |
| 47 | \( 1 + 35914993 T + p^{9} T^{2} \) |
| 53 | \( 1 + 66514064 T + p^{9} T^{2} \) |
| 59 | \( 1 - 108164002 T + p^{9} T^{2} \) |
| 61 | \( 1 + 207449912 T + p^{9} T^{2} \) |
| 67 | \( 1 + 193015514 T + p^{9} T^{2} \) |
| 71 | \( 1 - 201833497 T + p^{9} T^{2} \) |
| 73 | \( 1 - 121628110 T + p^{9} T^{2} \) |
| 79 | \( 1 - 112871912 T + p^{9} T^{2} \) |
| 83 | \( 1 + 308254212 T + p^{9} T^{2} \) |
| 89 | \( 1 - 6374870 T + p^{9} T^{2} \) |
| 97 | \( 1 + 871266886 T + p^{9} T^{2} \) |
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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
−9.451899056769897951568120158736, −8.511417017075618950220931610517, −8.080104505513244403896192203609, −6.78141102809020450958289041130, −5.65530512965511000019967477368, −4.89613701137359634104995728697, −3.13761692659489465309603798385, −1.88708775395415163012287723782, −1.72187108407626108074847091668, 0,
1.72187108407626108074847091668, 1.88708775395415163012287723782, 3.13761692659489465309603798385, 4.89613701137359634104995728697, 5.65530512965511000019967477368, 6.78141102809020450958289041130, 8.080104505513244403896192203609, 8.511417017075618950220931610517, 9.451899056769897951568120158736
