| 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 + 2.85e4·13-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 + 4.56e5·26-s − 2.53e6·27-s − 2.58e6·28-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 + 0.277·13-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 + 0.196·26-s − 0.916·27-s − 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{4} T \) |
| 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
−14.82018513514779883938816757446, −13.44736350363393059801394475741, −12.30808202967032660883118156330, −11.16718089565018451657033949052, −9.253269094196819735286777872271, −7.71431479179701964605801408256, −6.23085352644144134231919182517, −3.97115450154253890308563411468, −3.00825339531731739911485866313, 0,
3.00825339531731739911485866313, 3.97115450154253890308563411468, 6.23085352644144134231919182517, 7.71431479179701964605801408256, 9.253269094196819735286777872271, 11.16718089565018451657033949052, 12.30808202967032660883118156330, 13.44736350363393059801394475741, 14.82018513514779883938816757446
