L(s) = 1 | + (1.38 + 1.38i)2-s + (−2.07 − 2.07i)3-s + 1.84i·4-s + (−2.03 + 0.933i)5-s − 5.75i·6-s + (1.48 + 1.48i)7-s + (0.212 − 0.212i)8-s + 5.61i·9-s + (−4.11 − 1.52i)10-s + (3.83 − 3.83i)12-s + (1.72 − 1.72i)13-s + 4.12i·14-s + (6.15 + 2.27i)15-s + 4.28·16-s + (3.55 + 3.55i)17-s + (−7.78 + 7.78i)18-s + ⋯ |
L(s) = 1 | + (0.980 + 0.980i)2-s + (−1.19 − 1.19i)3-s + 0.923i·4-s + (−0.908 + 0.417i)5-s − 2.35i·6-s + (0.561 + 0.561i)7-s + (0.0750 − 0.0750i)8-s + 1.87i·9-s + (−1.30 − 0.481i)10-s + (1.10 − 1.10i)12-s + (0.477 − 0.477i)13-s + 1.10i·14-s + (1.58 + 0.588i)15-s + 1.07·16-s + (0.861 + 0.861i)17-s + (−1.83 + 1.83i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43489 + 0.691815i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43489 + 0.691815i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.03 - 0.933i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.38 - 1.38i)T + 2iT^{2} \) |
| 3 | \( 1 + (2.07 + 2.07i)T + 3iT^{2} \) |
| 7 | \( 1 + (-1.48 - 1.48i)T + 7iT^{2} \) |
| 13 | \( 1 + (-1.72 + 1.72i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.55 - 3.55i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.415T + 19T^{2} \) |
| 23 | \( 1 + (-4.64 - 4.64i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + (-0.431 + 0.431i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.329iT - 41T^{2} \) |
| 43 | \( 1 + (3.73 - 3.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.46 - 6.46i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.30 + 2.30i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.04iT - 59T^{2} \) |
| 61 | \( 1 + 1.43iT - 61T^{2} \) |
| 67 | \( 1 + (4.17 - 4.17i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.59T + 71T^{2} \) |
| 73 | \( 1 + (5.52 - 5.52i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.59T + 79T^{2} \) |
| 83 | \( 1 + (1.78 - 1.78i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.76iT - 89T^{2} \) |
| 97 | \( 1 + (-9.70 + 9.70i)T - 97iT^{2} \) |
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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
−11.19308024028300204036131833491, −10.28180130038283606672329024031, −8.310561071290929688070051240547, −7.78775455111922051563092441761, −6.94834734122985511237800049608, −6.23944613243983120562343248576, −5.48902953731211864497309592253, −4.70016328449199741715809548909, −3.30117967761616286616155832209, −1.26957639673263409539964319562,
0.962125355968218074130757121645, 3.19627594142506270861612309691, 4.10643164308798389609126381420, 4.75888343517502450493667541234, 5.23583041588581184417028585998, 6.62353102137168532571665893357, 7.924635334388883027102775681613, 9.065603662182040838669084117437, 10.28669794148567440091229636195, 10.74272564882502800516857631429