| L(s) = 1 | + (1.30 − 1.51i)2-s + (2.52 − 1.62i)3-s + (−0.569 − 3.95i)4-s + (−3.70 − 8.11i)5-s + (0.853 − 5.93i)6-s + (−24.4 + 7.16i)7-s + (−6.73 − 4.32i)8-s + (3.73 − 8.18i)9-s + (−17.1 − 5.02i)10-s + (−47.4 − 54.7i)11-s + (−7.85 − 9.06i)12-s + (64.0 + 18.8i)13-s + (−21.1 + 46.2i)14-s + (−22.5 − 14.4i)15-s + (−15.3 + 4.50i)16-s + (−6.65 + 46.3i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.331 − 0.725i)5-s + (0.0580 − 0.404i)6-s + (−1.31 + 0.386i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.541 − 0.158i)10-s + (−1.30 − 1.50i)11-s + (−0.189 − 0.218i)12-s + (1.36 + 0.401i)13-s + (−0.403 + 0.883i)14-s + (−0.387 − 0.248i)15-s + (−0.239 + 0.0704i)16-s + (−0.0950 + 0.660i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.290682 - 1.52086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.290682 - 1.52086i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 1.51i)T \) |
| 3 | \( 1 + (-2.52 + 1.62i)T \) |
| 23 | \( 1 + (-106. + 26.9i)T \) |
| good | 5 | \( 1 + (3.70 + 8.11i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (24.4 - 7.16i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (47.4 + 54.7i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-64.0 - 18.8i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (6.65 - 46.3i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (7.81 + 54.3i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-22.2 + 154. i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (97.8 + 62.8i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (172. - 377. i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (78.6 + 172. i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-242. + 155. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 - 46.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-735. + 215. i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-274. - 80.5i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (10.4 + 6.74i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-5.50 + 6.35i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-689. + 795. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-121. - 847. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (835. + 245. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (0.313 - 0.686i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (772. - 496. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (278. + 608. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−12.57373561484473364808204152350, −11.35996215519763624670888880276, −10.34495284288625791943211909669, −8.908809231512166286957090679555, −8.414881346974138034425766922259, −6.56598882528398543878527693980, −5.52979002400889818677536353739, −3.80044509103964662400956884147, −2.75367497100661103018764588885, −0.62077137558109252178290851773,
2.87616634279841569828544670432, 3.82140875916054913813559793011, 5.38277136281290393315648525513, 6.90209498423629164463718671515, 7.48889143860065126709638406081, 8.942187723972305198012959221912, 10.13247434261200527767507969607, 10.91343590564923446272150696692, 12.67801329422172306408049688699, 13.09647232118611514529445340195
