| L(s) = 1 | + (−1.06 + 0.140i)2-s + (−1.69 + 0.369i)3-s + (−0.818 + 0.219i)4-s + (−0.828 − 1.68i)5-s + (1.74 − 0.630i)6-s + (2.57 + 1.26i)7-s + (2.82 − 1.16i)8-s + (2.72 − 1.25i)9-s + (1.11 + 1.67i)10-s + (2.97 − 1.01i)11-s + (1.30 − 0.673i)12-s + (0.326 + 1.21i)13-s + (−2.91 − 0.989i)14-s + (2.02 + 2.53i)15-s + (−1.37 + 0.793i)16-s + (−3.15 − 2.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.752 + 0.0990i)2-s + (−0.976 + 0.213i)3-s + (−0.409 + 0.109i)4-s + (−0.370 − 0.751i)5-s + (0.714 − 0.257i)6-s + (0.972 + 0.479i)7-s + (0.998 − 0.413i)8-s + (0.908 − 0.416i)9-s + (0.353 + 0.528i)10-s + (0.897 − 0.304i)11-s + (0.376 − 0.194i)12-s + (0.0905 + 0.337i)13-s + (−0.779 − 0.264i)14-s + (0.522 + 0.655i)15-s + (−0.343 + 0.198i)16-s + (−0.764 − 0.644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.550113 - 0.0600882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.550113 - 0.0600882i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.69 - 0.369i)T \) |
| 17 | \( 1 + (3.15 + 2.65i)T \) |
| good | 2 | \( 1 + (1.06 - 0.140i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (0.828 + 1.68i)T + (-3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (-2.57 - 1.26i)T + (4.26 + 5.55i)T^{2} \) |
| 11 | \( 1 + (-2.97 + 1.01i)T + (8.72 - 6.69i)T^{2} \) |
| 13 | \( 1 + (-0.326 - 1.21i)T + (-11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-3.20 - 1.32i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-4.17 + 4.75i)T + (-3.00 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.202 - 3.09i)T + (-28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-3.39 + 9.99i)T + (-24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (-2.17 - 10.9i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-2.20 + 0.144i)T + (40.6 - 5.35i)T^{2} \) |
| 43 | \( 1 + (-0.443 - 0.340i)T + (11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (0.911 - 3.40i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.19 - 5.29i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-0.121 + 0.919i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-5.16 + 10.4i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (0.933 + 0.538i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.90 + 1.37i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (7.27 + 4.85i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.886 + 2.61i)T + (-62.6 + 48.0i)T^{2} \) |
| 83 | \( 1 + (-1.28 - 9.79i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (8.37 - 8.37i)T - 89iT^{2} \) |
| 97 | \( 1 + (-1.70 - 0.112i)T + (96.1 + 12.6i)T^{2} \) |
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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
−12.69580636154863568702092693058, −11.72596274512810848408295699327, −11.07786508473275755685985745628, −9.675477988052239564273586008520, −8.884674499232103553612859099720, −7.961560861941648382314283569171, −6.58265147658647194059186487034, −4.98339214589743436139500491695, −4.33353019361779765265311126659, −1.04137130197357097788111364959,
1.33223533153866278461061782610, 4.13139818677322741905774914320, 5.27881297647923277345829893801, 6.91179800659832464780744823002, 7.64575592758182432570705924396, 8.951940679572172336023859953839, 10.20100459768940595172124322020, 11.02117789681044932961712507282, 11.52881095291519474861444073149, 12.93847131764276776842252404896
