L(s) = 1 | + (0.456 − 0.456i)3-s + (0.456 − 2.18i)5-s + (2.79 + 2.79i)7-s + 2.58i·9-s − 4.37i·11-s + (−1.73 − 1.73i)13-s + (−0.791 − 1.20i)15-s + (3 − 3i)17-s + 3.46·19-s + 2.55·21-s + (0.791 − 0.791i)23-s + (−4.58 − 1.99i)25-s + (2.55 + 2.55i)27-s + 5.29i·29-s − 1.58i·31-s + ⋯ |
L(s) = 1 | + (0.263 − 0.263i)3-s + (0.204 − 0.978i)5-s + (1.05 + 1.05i)7-s + 0.860i·9-s − 1.31i·11-s + (−0.480 − 0.480i)13-s + (−0.204 − 0.312i)15-s + (0.727 − 0.727i)17-s + 0.794·19-s + 0.556·21-s + (0.164 − 0.164i)23-s + (−0.916 − 0.399i)25-s + (0.490 + 0.490i)27-s + 0.982i·29-s − 0.284i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.148671310\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.148671310\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.456 + 2.18i)T \) |
good | 3 | \( 1 + (-0.456 + 0.456i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.79 - 2.79i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.37iT - 11T^{2} \) |
| 13 | \( 1 + (1.73 + 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + (-0.791 + 0.791i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.29iT - 29T^{2} \) |
| 31 | \( 1 + 1.58iT - 31T^{2} \) |
| 37 | \( 1 + (5.19 - 5.19i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.58T + 41T^{2} \) |
| 43 | \( 1 + (-8.29 + 8.29i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.791 - 0.791i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.64 - 2.64i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 + 9.66T + 61T^{2} \) |
| 67 | \( 1 + (8.29 + 8.29i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (0.582 + 0.582i)T + 73iT^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (-4.83 + 4.83i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.16iT - 89T^{2} \) |
| 97 | \( 1 + (-0.582 + 0.582i)T - 97iT^{2} \) |
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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
−9.191793173131798170026398689841, −8.803891205178774141331437885052, −7.963255473107796436289823434514, −7.52568625546845865005178075160, −5.89933965757896409529912349584, −5.29109859685167791488360434252, −4.80991315529047471813027697613, −3.19322040151563065841524532809, −2.19673399087993089553681857409, −1.03134706760424732809863548509,
1.36404994139170662485891913191, 2.56866258472619128966459326508, 3.83898479143446412870558235589, 4.36759794133479329447175130670, 5.57153937644792407486502800265, 6.67543628423932612868921559487, 7.40695722356328192417620332022, 7.83858552861095712432919055794, 9.194358681474842354433933408693, 9.858837318533893795554051827755