L(s) = 1 | − 2.24·2-s + 3.03·4-s + 2.70·5-s − 1.81·7-s − 2.33·8-s − 6.08·10-s + 1.28·11-s + 4.53·13-s + 4.07·14-s − 0.841·16-s − 4.48·17-s + 2.67·19-s + 8.23·20-s − 2.89·22-s + 23-s + 2.33·25-s − 10.1·26-s − 5.51·28-s + 29-s − 5.79·31-s + 6.55·32-s + 10.0·34-s − 4.91·35-s − 5.91·37-s − 5.99·38-s − 6.31·40-s − 7.82·41-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 1.51·4-s + 1.21·5-s − 0.686·7-s − 0.824·8-s − 1.92·10-s + 0.388·11-s + 1.25·13-s + 1.08·14-s − 0.210·16-s − 1.08·17-s + 0.613·19-s + 1.84·20-s − 0.616·22-s + 0.208·23-s + 0.467·25-s − 1.99·26-s − 1.04·28-s + 0.185·29-s − 1.04·31-s + 1.15·32-s + 1.72·34-s − 0.831·35-s − 0.973·37-s − 0.973·38-s − 0.999·40-s − 1.22·41-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 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 4.48T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 + 7.87T + 59T^{2} \) |
| 61 | \( 1 - 0.0192T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 - 8.65T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 7.50T + 97T^{2} \) |
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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
−7.905868653418773184097557830094, −6.96872351534618296698441869540, −6.54661885085538429116882954472, −5.94676779093067338062372085369, −5.04662227909735942253323892764, −3.83170049517059467314550131931, −2.90324061930866283576997144192, −1.86516315426915866092447823202, −1.34287562312540155761585199622, 0,
1.34287562312540155761585199622, 1.86516315426915866092447823202, 2.90324061930866283576997144192, 3.83170049517059467314550131931, 5.04662227909735942253323892764, 5.94676779093067338062372085369, 6.54661885085538429116882954472, 6.96872351534618296698441869540, 7.905868653418773184097557830094