| L(s) = 1 | + 1.54·2-s − 2.94·3-s + 0.375·4-s − 5-s − 4.53·6-s + 7-s − 2.50·8-s + 5.66·9-s − 1.54·10-s + 2.12·11-s − 1.10·12-s + 1.14·13-s + 1.54·14-s + 2.94·15-s − 4.61·16-s + 0.557·17-s + 8.73·18-s − 0.916·19-s − 0.375·20-s − 2.94·21-s + 3.27·22-s − 9.06·23-s + 7.37·24-s + 25-s + 1.75·26-s − 7.85·27-s + 0.375·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s − 1.69·3-s + 0.187·4-s − 0.447·5-s − 1.85·6-s + 0.377·7-s − 0.885·8-s + 1.88·9-s − 0.487·10-s + 0.641·11-s − 0.319·12-s + 0.316·13-s + 0.411·14-s + 0.760·15-s − 1.15·16-s + 0.135·17-s + 2.05·18-s − 0.210·19-s − 0.0840·20-s − 0.642·21-s + 0.699·22-s − 1.89·23-s + 1.50·24-s + 0.200·25-s + 0.344·26-s − 1.51·27-s + 0.0710·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 + 2.94T + 3T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 - 0.557T + 17T^{2} \) |
| 19 | \( 1 + 0.916T + 19T^{2} \) |
| 23 | \( 1 + 9.06T + 23T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 - 5.30T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 + 2.50T + 43T^{2} \) |
| 47 | \( 1 + 1.54T + 47T^{2} \) |
| 53 | \( 1 - 2.18T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 6.16T + 61T^{2} \) |
| 67 | \( 1 - 0.219T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 2.54T + 83T^{2} \) |
| 89 | \( 1 - 1.54T + 89T^{2} \) |
| 97 | \( 1 - 5.82T + 97T^{2} \) |
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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
−7.13473320280829873319140530286, −6.38202402790223774250962312423, −6.04238284690857432491886436594, −5.34230246503293867175324160923, −4.73623436540848092804855639510, −4.08133430080055137786103422328, −3.65176873490820961614368669893, −2.26237630977274565471741302290, −1.05020836179119630370394967103, 0,
1.05020836179119630370394967103, 2.26237630977274565471741302290, 3.65176873490820961614368669893, 4.08133430080055137786103422328, 4.73623436540848092804855639510, 5.34230246503293867175324160923, 6.04238284690857432491886436594, 6.38202402790223774250962312423, 7.13473320280829873319140530286
