| L(s) = 1 | + (−0.453 − 0.891i)2-s + (0.322 − 2.03i)3-s + (−0.587 + 0.809i)4-s + (−1.79 − 1.33i)5-s + (−1.95 + 0.636i)6-s + (0.386 − 0.0611i)7-s + (0.987 + 0.156i)8-s + (−1.18 − 0.384i)9-s + (−0.373 + 2.20i)10-s + (−3.24 + 0.696i)11-s + (1.45 + 1.45i)12-s + (5.11 − 2.60i)13-s + (−0.229 − 0.316i)14-s + (−3.29 + 3.22i)15-s + (−0.309 − 0.951i)16-s + (0.440 + 0.224i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 − 0.630i)2-s + (0.186 − 1.17i)3-s + (−0.293 + 0.404i)4-s + (−0.802 − 0.596i)5-s + (−0.799 + 0.259i)6-s + (0.145 − 0.0231i)7-s + (0.349 + 0.0553i)8-s + (−0.394 − 0.128i)9-s + (−0.118 + 0.697i)10-s + (−0.977 + 0.210i)11-s + (0.420 + 0.420i)12-s + (1.41 − 0.723i)13-s + (−0.0614 − 0.0845i)14-s + (−0.850 + 0.831i)15-s + (−0.0772 − 0.237i)16-s + (0.106 + 0.0544i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.355182 - 0.745268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.355182 - 0.745268i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.453 + 0.891i)T \) |
| 5 | \( 1 + (1.79 + 1.33i)T \) |
| 11 | \( 1 + (3.24 - 0.696i)T \) |
| good | 3 | \( 1 + (-0.322 + 2.03i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (-0.386 + 0.0611i)T + (6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-5.11 + 2.60i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.440 - 0.224i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.61 + 1.90i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.05 + 1.05i)T - 23iT^{2} \) |
| 29 | \( 1 + (-6.85 - 4.97i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.58 - 4.87i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.45 - 9.20i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (3.50 + 4.82i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (3.86 + 3.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.76 + 0.596i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (0.830 + 1.62i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (1.77 - 2.44i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.86 + 2.55i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (10.6 + 10.6i)T + 67iT^{2} \) |
| 71 | \( 1 + (-3.83 - 11.8i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.89 - 11.9i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (2.96 - 9.12i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.69 + 5.29i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 - 9.83iT - 89T^{2} \) |
| 97 | \( 1 + (-4.79 + 2.44i)T + (57.0 - 78.4i)T^{2} \) |
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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
−13.02599423431834045503876218766, −12.41033232757416787063067274559, −11.36926284439333369033394158931, −10.30053712230434034939988621356, −8.558437719170440091344606144510, −8.087206889794234773354258785032, −6.91721791001739654631168202060, −5.01451373623015347336511191892, −3.18252937433021844495566737287, −1.19153030309280689771235093034,
3.46539519465211544068246343191, 4.64537779815049752486868191500, 6.17922187120906596518292648799, 7.64459072943214194545313325603, 8.593201830696596124442542255446, 9.792288699518951202882198719502, 10.72248600918111622190624489515, 11.56051707023758991091214338321, 13.32435613008494854541596273902, 14.42390825088742606189983837773
