| L(s) = 1 | + (0.191 − 2.19i)2-s + (0.699 + 1.58i)3-s + (−2.79 − 0.492i)4-s + (2.03 − 0.919i)5-s + (3.60 − 1.22i)6-s + (−0.000371 + 0.000530i)7-s + (−0.474 + 1.77i)8-s + (−2.02 + 2.21i)9-s + (−1.62 − 4.64i)10-s + (0.925 − 2.54i)11-s + (−1.17 − 4.76i)12-s + (−3.16 + 0.277i)13-s + (0.00109 + 0.000915i)14-s + (2.88 + 2.58i)15-s + (−1.53 − 0.559i)16-s + (1.76 + 6.58i)17-s + ⋯ |
| L(s) = 1 | + (0.135 − 1.54i)2-s + (0.403 + 0.914i)3-s + (−1.39 − 0.246i)4-s + (0.911 − 0.411i)5-s + (1.47 − 0.501i)6-s + (−0.000140 + 0.000200i)7-s + (−0.167 + 0.626i)8-s + (−0.673 + 0.738i)9-s + (−0.513 − 1.46i)10-s + (0.278 − 0.766i)11-s + (−0.338 − 1.37i)12-s + (−0.878 + 0.0768i)13-s + (0.000291 + 0.000244i)14-s + (0.744 + 0.667i)15-s + (−0.384 − 0.139i)16-s + (0.428 + 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04020 - 0.860484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04020 - 0.860484i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.699 - 1.58i)T \) |
| 5 | \( 1 + (-2.03 + 0.919i)T \) |
| good | 2 | \( 1 + (-0.191 + 2.19i)T + (-1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (0.000371 - 0.000530i)T + (-2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.925 + 2.54i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.16 - 0.277i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.76 - 6.58i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.52 - 0.879i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.18 - 2.93i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (-4.37 + 3.67i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.944 - 5.35i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.253 + 0.0678i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-7.56 + 9.01i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (7.88 - 3.67i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (0.576 + 0.403i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (5.73 + 5.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.11 + 0.768i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.14 + 6.48i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.216 - 2.47i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (10.4 + 6.02i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.40 - 0.375i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (3.36 + 4.00i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-15.9 - 1.39i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (2.99 + 5.18i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.55 - 7.62i)T + (-62.3 + 74.3i)T^{2} \) |
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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
−12.86014317839543144689968464817, −11.95256926147578573434191859447, −10.71408113641731177061902502497, −10.11964693555212940937581982410, −9.302880656279941671501556958888, −8.310482616074013287672378964566, −5.92881041301478209233101911381, −4.59263758863996775818021154372, −3.40423922605923942236414821412, −1.97467639001751482865963233759,
2.46694476032567618355301476875, 4.89683178872033595091664962094, 6.14050967788560874477338006486, 6.98172011197973042523911694420, 7.68977971787775991638102693882, 8.991920159326819156178265957927, 9.908620716056712040590893326937, 11.74229335676859062885737876006, 12.89744473666371341762393656686, 13.83058296519437135416827162898
