| L(s) = 1 | − 75·3-s − 1.97e3·5-s + 1.01e4·7-s − 1.40e4·9-s − 1.88e4·11-s + 2.85e4·13-s + 1.48e5·15-s − 1.42e5·17-s − 8.33e4·19-s − 7.58e5·21-s + 5.36e5·23-s + 1.96e6·25-s + 2.53e6·27-s − 2.60e6·29-s + 2.21e6·31-s + 1.41e6·33-s − 2.00e7·35-s + 1.80e7·37-s − 2.14e6·39-s + 2.68e7·41-s + 4.22e7·43-s + 2.78e7·45-s − 3.59e7·47-s + 6.19e7·49-s + 1.06e7·51-s − 6.65e7·53-s + 3.73e7·55-s + ⋯ |
| L(s) = 1 | − 0.534·3-s − 1.41·5-s + 1.59·7-s − 0.714·9-s − 0.388·11-s + 0.277·13-s + 0.757·15-s − 0.413·17-s − 0.146·19-s − 0.851·21-s + 0.399·23-s + 1.00·25-s + 0.916·27-s − 0.682·29-s + 0.430·31-s + 0.207·33-s − 2.25·35-s + 1.58·37-s − 0.148·39-s + 1.48·41-s + 1.88·43-s + 1.01·45-s − 1.07·47-s + 1.53·49-s + 0.221·51-s − 1.15·53-s + 0.549·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - p^{4} T \) |
| 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
−10.87164736831376730501716357458, −9.014785991700314603661867409573, −8.040800780234869749936827364899, −7.56729365417839967163926607294, −6.03394843334441687020358376790, −4.88007654041583338667086649487, −4.14163118128551085213786457288, −2.65656802017290477783966125489, −1.08806307875591839704088806031, 0,
1.08806307875591839704088806031, 2.65656802017290477783966125489, 4.14163118128551085213786457288, 4.88007654041583338667086649487, 6.03394843334441687020358376790, 7.56729365417839967163926607294, 8.040800780234869749936827364899, 9.014785991700314603661867409573, 10.87164736831376730501716357458
