| L(s) = 1 | + (0.503 + 0.872i)2-s + (−0.352 + 0.295i)3-s + (0.492 − 0.853i)4-s + (0.517 + 2.93i)5-s + (−0.435 − 0.158i)6-s + (−0.307 − 1.74i)7-s + 3.00·8-s + (−0.484 + 2.74i)9-s + (−2.30 + 1.93i)10-s + (−1.58 − 1.32i)11-s + (0.0786 + 0.445i)12-s + (−0.513 − 2.91i)13-s + (1.36 − 1.14i)14-s + (−1.04 − 0.880i)15-s + (0.529 + 0.916i)16-s + (−0.859 − 1.48i)17-s + ⋯ |
| L(s) = 1 | + (0.356 + 0.616i)2-s + (−0.203 + 0.170i)3-s + (0.246 − 0.426i)4-s + (0.231 + 1.31i)5-s + (−0.177 − 0.0646i)6-s + (−0.116 − 0.659i)7-s + 1.06·8-s + (−0.161 + 0.915i)9-s + (−0.727 + 0.610i)10-s + (−0.477 − 0.400i)11-s + (0.0227 + 0.128i)12-s + (−0.142 − 0.807i)13-s + (0.365 − 0.306i)14-s + (−0.271 − 0.227i)15-s + (0.132 + 0.229i)16-s + (−0.208 − 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10786 + 0.565763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10786 + 0.565763i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-4.58 + 9.37i)T \) |
| good | 2 | \( 1 + (-0.503 - 0.872i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.352 - 0.295i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.517 - 2.93i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.307 + 1.74i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.58 + 1.32i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.513 + 2.91i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.859 + 1.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.50 + 4.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.38 - 2.39i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.16 + 2.65i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.604 + 3.43i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.89 - 10.7i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + 8.90T + 41T^{2} \) |
| 43 | \( 1 + (-3.82 - 6.62i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.40 - 0.876i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.78 - 10.1i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.27 - 6.10i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (11.1 + 4.04i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.94 - 11.0i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.48 + 9.49i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.91 - 4.12i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.26 + 7.18i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.45 + 2.06i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.70 - 2.43i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.11 + 6.33i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.68857842453865063380166410317, −13.54003639352800294013566772420, −11.37769356987933400214825516963, −10.60237355035279265250823157023, −10.09633362350527272156417223281, −7.946188514392535578811845661233, −6.99560157716682188112922625023, −6.00189216914133504407838648738, −4.76011827587760313252879124348, −2.71798376225856284794823555020,
1.96799890622564909836544745376, 3.90442605229413109277678770482, 5.23020802822857829495189884820, 6.68857130508922038611872276889, 8.296491170478097323475389981105, 9.166811246446935562187627778611, 10.53296111863206474054881920505, 12.04438715499501294377856750150, 12.29872893923455696946833023434, 13.04129213032339447197590553914
