| L(s) = 1 | + (1.45 − 0.256i)2-s + (−1.59 − 1.90i)3-s + (0.176 − 0.0642i)4-s + (0.658 − 2.13i)5-s + (−2.81 − 2.36i)6-s + (2.81 + 1.62i)7-s + (−2.32 + 1.34i)8-s + (−0.549 + 3.11i)9-s + (0.410 − 3.28i)10-s + (2.09 + 3.62i)11-s + (−0.404 − 0.233i)12-s + (1.14 − 1.36i)13-s + (4.51 + 1.64i)14-s + (−5.11 + 2.15i)15-s + (−3.32 + 2.79i)16-s + (−6.23 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (1.03 − 0.181i)2-s + (−0.921 − 1.09i)3-s + (0.0883 − 0.0321i)4-s + (0.294 − 0.955i)5-s + (−1.14 − 0.963i)6-s + (1.06 + 0.614i)7-s + (−0.820 + 0.473i)8-s + (−0.183 + 1.03i)9-s + (0.129 − 1.03i)10-s + (0.630 + 1.09i)11-s + (−0.116 − 0.0673i)12-s + (0.318 − 0.379i)13-s + (1.20 + 0.439i)14-s + (−1.32 + 0.557i)15-s + (−0.831 + 0.697i)16-s + (−1.51 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04953 - 0.683475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04953 - 0.683475i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.658 + 2.13i)T \) |
| 19 | \( 1 + (-4.09 + 1.49i)T \) |
| good | 2 | \( 1 + (-1.45 + 0.256i)T + (1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (1.59 + 1.90i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (-2.81 - 1.62i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.09 - 3.62i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.14 + 1.36i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.23 - 1.09i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.490 - 1.34i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0589 + 0.334i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.38 + 2.40i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.70iT - 37T^{2} \) |
| 41 | \( 1 + (5.46 - 4.58i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.38 - 9.29i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.438 + 0.0773i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (2.47 + 6.80i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.545 + 3.09i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.88 + 1.04i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-8.42 - 1.48i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (12.1 + 4.40i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.17 - 1.39i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (0.535 - 0.449i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.478 + 0.276i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.23 + 4.38i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-13.0 + 2.29i)T + (91.1 - 33.1i)T^{2} \) |
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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
−13.29982276507618406865121514417, −12.87735162187193078195072456296, −11.81711083193540393122411456907, −11.47403759787510355118685029025, −9.290452535172336242509330600991, −8.122329621420413423394407280572, −6.51667459186000841284198576728, −5.36905848801865434833442058861, −4.56987795486181824661303660507, −1.82137324990838993334138122889,
3.63380826760366207635408809333, 4.66632448902556127107575467227, 5.75940246324452233608639221663, 6.81949984157408828171355947821, 8.902709175870794215893044476455, 10.24444697306349787914751415169, 11.18034788694087811190339194326, 11.71409192543115686977852522864, 13.70272155319164935146049812595, 14.03296113526007883231948614600
