| L(s) = 1 | − 2.43·2-s + 3.92·4-s − 3.49·5-s − 4.87·7-s − 4.69·8-s + 8.51·10-s + 1.63·11-s − 5.07·13-s + 11.8·14-s + 3.58·16-s − 7.41·17-s + 0.289·19-s − 13.7·20-s − 3.97·22-s + 23-s + 7.21·25-s + 12.3·26-s − 19.1·28-s − 29-s − 1.20·31-s + 0.672·32-s + 18.0·34-s + 17.0·35-s + 2.03·37-s − 0.706·38-s + 16.4·40-s + 10.9·41-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.96·4-s − 1.56·5-s − 1.84·7-s − 1.66·8-s + 2.69·10-s + 0.491·11-s − 1.40·13-s + 3.17·14-s + 0.895·16-s − 1.79·17-s + 0.0665·19-s − 3.07·20-s − 0.846·22-s + 0.208·23-s + 1.44·25-s + 2.42·26-s − 3.61·28-s − 0.185·29-s − 0.217·31-s + 0.118·32-s + 3.09·34-s + 2.87·35-s + 0.333·37-s − 0.114·38-s + 2.59·40-s + 1.70·41-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 | \( 1 + 2.43T + 2T^{2} \) |
| 5 | \( 1 + 3.49T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 - 1.63T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 + 7.41T + 17T^{2} \) |
| 19 | \( 1 - 0.289T + 19T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 - 2.03T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.66T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 + 7.16T + 53T^{2} \) |
| 59 | \( 1 - 0.758T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 + 8.89T + 71T^{2} \) |
| 73 | \( 1 + 2.62T + 73T^{2} \) |
| 79 | \( 1 + 3.77T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
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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
−7.70588519264730892141130920715, −7.17452012920461145508247201767, −6.77867419146511994445684562658, −6.08419559910863459417612901598, −4.61406416098984487542357112045, −3.90252511074564357718859278305, −2.94681045564334834896007627645, −2.28695133763499289285849756069, −0.64327447369707838028740042386, 0,
0.64327447369707838028740042386, 2.28695133763499289285849756069, 2.94681045564334834896007627645, 3.90252511074564357718859278305, 4.61406416098984487542357112045, 6.08419559910863459417612901598, 6.77867419146511994445684562658, 7.17452012920461145508247201767, 7.70588519264730892141130920715
