| L(s) = 1 | + (−21.2 + 36.7i)2-s + (86.0 + 149. i)3-s + (−644. − 1.11e3i)4-s + (1.13e3 + 1.97e3i)5-s − 7.30e3·6-s + (4.12e3 + 7.14e3i)7-s + 3.29e4·8-s + (−4.96e3 + 8.60e3i)9-s − 9.66e4·10-s − 8.84e4·11-s + (1.10e5 − 1.92e5i)12-s + (3.52e4 + 6.11e4i)13-s − 3.49e5·14-s + (−1.96e5 + 3.39e5i)15-s + (−3.69e5 + 6.39e5i)16-s + (4.28e4 − 7.42e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.937 + 1.62i)2-s + (0.613 + 1.06i)3-s + (−1.25 − 2.17i)4-s + (0.815 + 1.41i)5-s − 2.30·6-s + (0.649 + 1.12i)7-s + 2.84·8-s + (−0.252 + 0.437i)9-s − 3.05·10-s − 1.82·11-s + (1.54 − 2.67i)12-s + (0.342 + 0.593i)13-s − 2.43·14-s + (−1.00 + 1.73i)15-s + (−1.40 + 2.43i)16-s + (0.124 − 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00626 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.00626 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.969548 - 0.975639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.969548 - 0.975639i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.06e7 + 4.12e6i)T \) |
| good | 2 | \( 1 + (21.2 - 36.7i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-86.0 - 149. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.13e3 - 1.97e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-4.12e3 - 7.14e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + 8.84e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-3.52e4 - 6.11e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-4.28e4 + 7.42e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-3.74e4 - 6.47e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 - 2.92e3T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.94e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.32e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (1.31e7 + 2.28e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 - 3.40e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (2.79e7 - 4.83e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-9.20e5 + 1.59e6i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (3.01e7 + 5.23e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-3.06e7 - 5.31e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (3.39e7 + 5.88e7i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 1.03e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-1.21e8 - 2.10e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.43e8 + 4.21e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-4.48e8 + 7.77e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 3.06e8T + 7.60e17T^{2} \) |
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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
−15.41068837483883700869714780199, −14.61607994994213791231344727858, −13.83222280899560666065727574484, −10.73691748491702409422599108412, −9.931509946693969500465527912771, −8.906672394397294148708662281855, −7.73650154094813502539959083098, −6.20036441620987530413705850540, −5.08148610956907271363323921229, −2.49344975519403509277328200124,
0.69402860158210561346809755278, 1.42012069966680954798395237883, 2.63813346708746732732215983961, 4.74772137425387251034205522755, 7.935569287609795704833953260143, 8.241382981881938067485293016221, 9.817056302954321336156166779313, 10.80098714472683697962874053642, 12.45338823845178205185075917833, 13.27110866144731991092790774194
