| L(s) = 1 | + 2.82i·3-s − 5.00·9-s + 4·11-s − 4.24i·13-s + 1.41i·17-s − 2.82·19-s + 4i·23-s − 5.65i·27-s − 8·29-s + 11.3i·33-s − 8i·37-s + 12·39-s − 7.07·41-s + 4i·43-s + 5.65i·47-s + ⋯ |
| L(s) = 1 | + 1.63i·3-s − 1.66·9-s + 1.20·11-s − 1.17i·13-s + 0.342i·17-s − 0.648·19-s + 0.834i·23-s − 1.08i·27-s − 1.48·29-s + 1.96i·33-s − 1.31i·37-s + 1.92·39-s − 1.10·41-s + 0.609i·43-s + 0.825i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3131133548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3131133548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.82iT - 3T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 5.65iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 14.1iT - 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 1.41iT - 97T^{2} \) |
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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
−8.942949144713583328654271604924, −8.220167574700814972990053722298, −7.41185165531857206164805955055, −6.36707529886502573121713057223, −5.65753792160790125380238052392, −5.06976062792617716273536412873, −4.09728979793307579763019489131, −3.72010209563036582183174753313, −2.92175210995622052799884762283, −1.57704306069223156705182802513,
0.080210690097360517716299427673, 1.44360355215686755567363126337, 1.87788481119226131117908204868, 2.93852520467237842583407740284, 3.99692838396465194262923819338, 4.81463451341051355008519600060, 6.04994341538371532034286777114, 6.39026765219186827450966096276, 7.07249938614242741938636024113, 7.52548814642455141748723577550
