L(s) = 1 | + (5.44 + 3.14i)2-s + (−10.4 − 18.1i)3-s + (3.74 + 6.49i)4-s + (−46.0 + 26.6i)5-s − 131. i·6-s + (−56.9 − 98.6i)7-s − 154. i·8-s + (−98.1 + 170. i)9-s − 334.·10-s − 271.·11-s + (78.5 − 136. i)12-s + (229. − 132. i)13-s − 715. i·14-s + (965. + 557. i)15-s + (603. − 1.04e3i)16-s + (961. + 554. i)17-s + ⋯ |
L(s) = 1 | + (0.962 + 0.555i)2-s + (−0.672 − 1.16i)3-s + (0.117 + 0.202i)4-s + (−0.824 + 0.475i)5-s − 1.49i·6-s + (−0.439 − 0.760i)7-s − 0.850i·8-s + (−0.404 + 0.699i)9-s − 1.05·10-s − 0.677·11-s + (0.157 − 0.272i)12-s + (0.376 − 0.217i)13-s − 0.975i·14-s + (1.10 + 0.639i)15-s + (0.589 − 1.02i)16-s + (0.806 + 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.595247 - 1.02764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.595247 - 1.02764i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-6.09e3 - 5.67e3i)T \) |
good | 2 | \( 1 + (-5.44 - 3.14i)T + (16 + 27.7i)T^{2} \) |
| 3 | \( 1 + (10.4 + 18.1i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (46.0 - 26.6i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (56.9 + 98.6i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + 271.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-229. + 132. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-961. - 554. i)T + (7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-576. + 332. i)T + (1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + 991. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.70e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 9.34e3iT - 2.86e7T^{2} \) |
| 41 | \( 1 + (8.72e3 + 1.51e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + 1.67e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.18e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.12e4 - 1.94e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-3.02e4 - 1.74e4i)T + (3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-4.78e4 + 2.76e4i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-7.55e3 - 1.30e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-1.01e3 - 1.75e3i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + 7.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.73e4 - 1.00e4i)T + (1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.38e4 + 5.86e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-3.33e4 - 1.92e4i)T + (2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 4.56e4iT - 8.58e9T^{2} \) |
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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
−14.89179247084102405525637256316, −13.54560898145369547835315351780, −12.87033660476690624989460596421, −11.70558457096928780064112667806, −10.28775329116962743365566800128, −7.67363652971292954213011155252, −6.85780005219609559930302916518, −5.63363264234079921188526684957, −3.72074870849513995346524285646, −0.55663310693845561376554096770,
3.27846939406728720513866471204, 4.60578744932613501412898826244, 5.59101281091913152238655725237, 8.232014468179060515034034352872, 9.817405643590394068504744481482, 11.27365359303593891829776912815, 11.99934042952069816677216774788, 13.07880105736769078133566665305, 14.65021615034432596491349398611, 15.91457049399617068038383593344