| L(s) = 1 | − 0.224·2-s − 0.437·3-s − 1.94·4-s − 5-s + 0.0983·6-s − 7-s + 0.888·8-s − 2.80·9-s + 0.224·10-s − 2.09·11-s + 0.852·12-s + 0.534·13-s + 0.224·14-s + 0.437·15-s + 3.69·16-s − 6.23·17-s + 0.631·18-s + 1.07·19-s + 1.94·20-s + 0.437·21-s + 0.471·22-s − 1.32·23-s − 0.388·24-s + 25-s − 0.120·26-s + 2.54·27-s + 1.94·28-s + ⋯ |
| L(s) = 1 | − 0.159·2-s − 0.252·3-s − 0.974·4-s − 0.447·5-s + 0.0401·6-s − 0.377·7-s + 0.313·8-s − 0.936·9-s + 0.0711·10-s − 0.632·11-s + 0.246·12-s + 0.148·13-s + 0.0600·14-s + 0.112·15-s + 0.924·16-s − 1.51·17-s + 0.148·18-s + 0.246·19-s + 0.435·20-s + 0.0954·21-s + 0.100·22-s − 0.275·23-s − 0.0793·24-s + 0.200·25-s − 0.0235·26-s + 0.489·27-s + 0.368·28-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 | \( 1 + 0.224T + 2T^{2} \) |
| 3 | \( 1 + 0.437T + 3T^{2} \) |
| 11 | \( 1 + 2.09T + 11T^{2} \) |
| 13 | \( 1 - 0.534T + 13T^{2} \) |
| 17 | \( 1 + 6.23T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 0.363T + 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 53 | \( 1 + 1.21T + 53T^{2} \) |
| 59 | \( 1 - 3.69T + 59T^{2} \) |
| 61 | \( 1 - 8.13T + 61T^{2} \) |
| 67 | \( 1 + 7.21T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 + 1.62T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 + 4.04T + 97T^{2} \) |
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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
−7.66880668145314054500443526447, −6.79944446920424149502086286806, −5.95445440765104532666528857405, −5.47107067042876012286467373066, −4.50408414891418315508109029421, −4.14398694789400299113675846584, −3.07924430493335289233111464214, −2.38756081160407878397135283016, −0.836804254196414054295350422453, 0,
0.836804254196414054295350422453, 2.38756081160407878397135283016, 3.07924430493335289233111464214, 4.14398694789400299113675846584, 4.50408414891418315508109029421, 5.47107067042876012286467373066, 5.95445440765104532666528857405, 6.79944446920424149502086286806, 7.66880668145314054500443526447
