| L(s) = 1 | + (−11.9 + 20.6i)2-s + (13.2 + 22.9i)3-s + (−28.5 − 49.3i)4-s + (−269. − 466. i)5-s − 630.·6-s + (2.39e3 + 4.14e3i)7-s − 1.08e4·8-s + (9.49e3 − 1.64e4i)9-s + 1.28e4·10-s − 5.09e4·11-s + (753. − 1.30e3i)12-s + (−5.88e4 − 1.01e5i)13-s − 1.14e5·14-s + (7.12e3 − 1.23e4i)15-s + (1.44e5 − 2.49e5i)16-s + (−7.77e4 + 1.34e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.527 + 0.912i)2-s + (0.0942 + 0.163i)3-s + (−0.0556 − 0.0964i)4-s + (−0.192 − 0.333i)5-s − 0.198·6-s + (0.376 + 0.652i)7-s − 0.936·8-s + (0.482 − 0.835i)9-s + 0.406·10-s − 1.04·11-s + (0.0104 − 0.0181i)12-s + (−0.571 − 0.989i)13-s − 0.794·14-s + (0.0363 − 0.0629i)15-s + (0.549 − 0.951i)16-s + (−0.225 + 0.391i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.712080 - 0.248531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.712080 - 0.248531i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-3.31e6 + 1.09e7i)T \) |
| good | 2 | \( 1 + (11.9 - 20.6i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-13.2 - 22.9i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (269. + 466. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-2.39e3 - 4.14e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + 5.09e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (5.88e4 + 1.01e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (7.77e4 - 1.34e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.91e5 + 3.32e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 - 4.38e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.49e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.77e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (-1.94e6 - 3.36e6i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + 3.51e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.63e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-2.91e7 + 5.05e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (5.71e7 - 9.89e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-9.05e6 - 1.56e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (9.59e7 + 1.66e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.25e8 + 2.17e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 1.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (1.59e8 + 2.76e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-5.04e7 + 8.73e7i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (5.07e8 - 8.78e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 5.00e8T + 7.60e17T^{2} \) |
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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
−14.94319875965072355706477008892, −12.90368683170925282188812334537, −11.99294973860137564886144907381, −10.20226520613947080021960277413, −8.783919889745215183081074410051, −7.941249593108414958462292013742, −6.50924732793038938222114344955, −4.98898576918468437945406644529, −2.84473327601138749345009442245, −0.32955053623997700340164217702,
1.43850208566497061440691640146, 2.74755884246997370022198301397, 4.76285079223014687234663752742, 6.89244632490485481691419489677, 8.227754941844520123841209029009, 9.920391302705850937406613519890, 10.70427957873830486804615663371, 11.73597358656968272852612514766, 13.14693344303550987254936951322, 14.37631555738074206054445947183
