| L(s) = 1 | + 2.51·2-s + 1.25·3-s + 4.31·4-s + 5-s + 3.15·6-s + 1.61·7-s + 5.80·8-s − 1.42·9-s + 2.51·10-s + 3.28·11-s + 5.40·12-s − 6.35·13-s + 4.05·14-s + 1.25·15-s + 5.96·16-s − 3.58·18-s − 0.747·19-s + 4.31·20-s + 2.02·21-s + 8.25·22-s − 0.143·23-s + 7.28·24-s + 25-s − 15.9·26-s − 5.55·27-s + 6.96·28-s + 8.80·29-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.724·3-s + 2.15·4-s + 0.447·5-s + 1.28·6-s + 0.610·7-s + 2.05·8-s − 0.475·9-s + 0.794·10-s + 0.990·11-s + 1.56·12-s − 1.76·13-s + 1.08·14-s + 0.323·15-s + 1.49·16-s − 0.844·18-s − 0.171·19-s + 0.964·20-s + 0.442·21-s + 1.76·22-s − 0.0298·23-s + 1.48·24-s + 0.200·25-s − 3.13·26-s − 1.06·27-s + 1.31·28-s + 1.63·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.494521329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.494521329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 - 1.25T + 3T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 3.28T + 11T^{2} \) |
| 13 | \( 1 + 6.35T + 13T^{2} \) |
| 19 | \( 1 + 0.747T + 19T^{2} \) |
| 23 | \( 1 + 0.143T + 23T^{2} \) |
| 29 | \( 1 - 8.80T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 - 0.621T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 0.464T + 61T^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 - 4.25T + 71T^{2} \) |
| 73 | \( 1 + 1.21T + 73T^{2} \) |
| 79 | \( 1 - 5.04T + 79T^{2} \) |
| 83 | \( 1 + 3.58T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 6.96T + 97T^{2} \) |
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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
−9.516133477878105477568880358201, −8.691649066136253688059565302703, −7.63886949275120348369129717056, −6.90072599984722610196515921719, −6.03852194658805362400840550326, −5.14149760331335494837069321185, −4.53545690787018558049178353788, −3.50966834433199825768572092991, −2.62224197978066023265187732722, −1.87575599775952143264366509055,
1.87575599775952143264366509055, 2.62224197978066023265187732722, 3.50966834433199825768572092991, 4.53545690787018558049178353788, 5.14149760331335494837069321185, 6.03852194658805362400840550326, 6.90072599984722610196515921719, 7.63886949275120348369129717056, 8.691649066136253688059565302703, 9.516133477878105477568880358201
