| L(s) = 1 | + (−0.304 + 0.0988i)2-s + (−1.70 + 0.303i)3-s + (−1.53 + 1.11i)4-s + (−2.00 − 1.15i)5-s + (0.488 − 0.260i)6-s + (−2.85 − 1.27i)7-s + (0.732 − 1.00i)8-s + (2.81 − 1.03i)9-s + (0.725 + 0.154i)10-s + (0.566 + 5.38i)11-s + (2.27 − 2.36i)12-s + (−3.21 + 2.89i)13-s + (0.995 + 0.104i)14-s + (3.77 + 1.36i)15-s + (1.04 − 3.23i)16-s + (0.332 − 3.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.215 + 0.0699i)2-s + (−0.984 + 0.175i)3-s + (−0.767 + 0.557i)4-s + (−0.897 − 0.518i)5-s + (0.199 − 0.106i)6-s + (−1.08 − 0.481i)7-s + (0.259 − 0.356i)8-s + (0.938 − 0.345i)9-s + (0.229 + 0.0487i)10-s + (0.170 + 1.62i)11-s + (0.657 − 0.683i)12-s + (−0.891 + 0.802i)13-s + (0.266 + 0.0279i)14-s + (0.974 + 0.352i)15-s + (0.262 − 0.807i)16-s + (0.0805 − 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000486582 + 0.0765252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000486582 + 0.0765252i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.70 - 0.303i)T \) |
| 31 | \( 1 + (5.14 - 2.11i)T \) |
| good | 2 | \( 1 + (0.304 - 0.0988i)T + (1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (2.00 + 1.15i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.85 + 1.27i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.566 - 5.38i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (3.21 - 2.89i)T + (1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.332 + 3.16i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (1.18 - 1.32i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.927 + 0.674i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.38 + 4.26i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.91 + 1.68i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.17 - 5.51i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-6.01 - 5.41i)T + (4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (11.6 + 3.77i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.14 - 0.953i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (0.165 + 0.776i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 4.60iT - 61T^{2} \) |
| 67 | \( 1 + (-0.815 + 1.41i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.94 - 11.1i)T + (-47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (1.60 - 0.168i)T + (71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (6.78 + 0.713i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-9.11 - 1.93i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-7.04 + 5.12i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.99 - 1.45i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.67225589738829577130969469260, −13.04793432874498300517161457665, −12.43767292817867691835812998220, −11.69075534440077245574479239124, −9.910887189676900524245915140178, −9.459751338917327066835685468422, −7.65661860982335542995176404101, −6.82738623236100214570220308673, −4.75629143233784452021713591361, −4.04427374140140567549133046372,
0.11018111570256250771569508576, 3.60046229413752448854655343912, 5.39423639301292736491798142010, 6.34878544281314072814301345460, 7.86446693880566387427166867835, 9.247354245833318150101301609183, 10.46909882016200845897333863999, 11.20043303995458104989617719987, 12.46393616933118752199503165680, 13.28144323919369555484863327656
