| L(s) = 1 | + (−5.51 − 5.51i)2-s + 143. i·3-s − 195. i·4-s + (357. − 357. i)5-s + (792. − 792. i)6-s − 623.·7-s + (−2.48e3 + 2.48e3i)8-s − 1.40e4·9-s − 3.94e3·10-s − 1.14e4i·11-s + 2.80e4·12-s + (1.48e4 − 1.48e4i)13-s + (3.44e3 + 3.44e3i)14-s + (5.14e4 + 5.14e4i)15-s − 2.25e4·16-s + (9.38e3 − 9.38e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.344 − 0.344i)2-s + 1.77i·3-s − 0.762i·4-s + (0.572 − 0.572i)5-s + (0.611 − 0.611i)6-s − 0.259·7-s + (−0.607 + 0.607i)8-s − 2.14·9-s − 0.394·10-s − 0.782i·11-s + 1.35·12-s + (0.520 − 0.520i)13-s + (0.0895 + 0.0895i)14-s + (1.01 + 1.01i)15-s − 0.343·16-s + (0.112 − 0.112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.15958 - 0.625210i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15958 - 0.625210i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.79e6 - 5.29e5i)T \) |
| good | 2 | \( 1 + (5.51 + 5.51i)T + 256iT^{2} \) |
| 3 | \( 1 - 143. iT - 6.56e3T^{2} \) |
| 5 | \( 1 + (-357. + 357. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + 623.T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.14e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (-1.48e4 + 1.48e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (-9.38e3 + 9.38e3i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (-1.51e5 + 1.51e5i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 + (-2.89e5 + 2.89e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (-3.89e4 - 3.89e4i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 + (5.59e5 + 5.59e5i)T + 8.52e11iT^{2} \) |
| 41 | \( 1 + 4.22e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (-8.64e5 + 8.64e5i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 - 5.06e5T + 2.38e13T^{2} \) |
| 53 | \( 1 + 6.50e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.00e5 + 4.00e5i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (4.01e6 + 4.01e6i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 - 2.61e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 4.54e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 4.65e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (-2.23e7 + 2.23e7i)T - 1.51e15iT^{2} \) |
| 83 | \( 1 + 2.48e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-6.18e7 - 6.18e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + (-8.56e7 + 8.56e7i)T - 7.83e15iT^{2} \) |
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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
−14.67133566244169369533799533238, −13.45207781869783940717884121423, −11.32121086683632041072940372234, −10.55474028711117326813765969432, −9.417317376990978907348119442703, −8.883251654667394629487526753181, −5.81121855992574105764690710907, −4.91608576609537382774119718731, −3.04730649926753154488678819370, −0.62931255392433665291944219954,
1.48064435562478619459791173471, 3.03094847139391425316569570077, 6.12646007617013878262977733412, 7.08217833596354130095371911763, 7.944239317553439551451464149354, 9.457416596660148956555546315848, 11.52784341237982829365769386272, 12.56007058058475299662863651715, 13.40628759980641866640998315007, 14.49178195999396168332603395880
