| L(s) = 1 | − 2.24·2-s + 3.03·4-s − 3.33·5-s + 0.336·7-s − 2.31·8-s + 7.47·10-s − 5.35·11-s − 2.46·13-s − 0.754·14-s − 0.877·16-s − 3.61·17-s − 5.53·19-s − 10.0·20-s + 12.0·22-s − 23-s + 6.10·25-s + 5.53·26-s + 1.01·28-s + 29-s + 8.47·31-s + 6.59·32-s + 8.10·34-s − 1.12·35-s + 9.55·37-s + 12.4·38-s + 7.70·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.51·4-s − 1.49·5-s + 0.127·7-s − 0.817·8-s + 2.36·10-s − 1.61·11-s − 0.684·13-s − 0.201·14-s − 0.219·16-s − 0.876·17-s − 1.27·19-s − 2.25·20-s + 2.56·22-s − 0.208·23-s + 1.22·25-s + 1.08·26-s + 0.192·28-s + 0.185·29-s + 1.52·31-s + 1.16·32-s + 1.38·34-s − 0.189·35-s + 1.57·37-s + 2.01·38-s + 1.21·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 7 | \( 1 - 0.336T + 7T^{2} \) |
| 11 | \( 1 + 5.35T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 - 9.57T + 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 - 1.21T + 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 - 5.16T + 67T^{2} \) |
| 71 | \( 1 + 2.09T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 3.01T + 97T^{2} \) |
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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
−7.88003633310572652608040855165, −7.41386514891966927120948919249, −6.74973234567578877527202425680, −5.77980557314294829302638463804, −4.47070354499672244849869031776, −4.32204985951967324656079172140, −2.70250782972099157010735829662, −2.37562826492444915867159441044, −0.77069239605553558716634987342, 0,
0.77069239605553558716634987342, 2.37562826492444915867159441044, 2.70250782972099157010735829662, 4.32204985951967324656079172140, 4.47070354499672244849869031776, 5.77980557314294829302638463804, 6.74973234567578877527202425680, 7.41386514891966927120948919249, 7.88003633310572652608040855165
