| L(s) = 1 | + 2.41·3-s + (1.94 + 1.10i)5-s − i·7-s + 2.84·9-s − 2.55i·11-s + 5.47·13-s + (4.70 + 2.66i)15-s − 2.44i·17-s − 4.99i·19-s − 2.41i·21-s + 1.10i·23-s + (2.56 + 4.29i)25-s − 0.376·27-s − 1.96i·29-s − 2.33·31-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + (0.869 + 0.493i)5-s − 0.377i·7-s + 0.948·9-s − 0.770i·11-s + 1.51·13-s + (1.21 + 0.688i)15-s − 0.593i·17-s − 1.14i·19-s − 0.527i·21-s + 0.229i·23-s + (0.513 + 0.858i)25-s − 0.0724·27-s − 0.364i·29-s − 0.419·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.643179193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.643179193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.94 - 1.10i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 11 | \( 1 + 2.55iT - 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 2.44iT - 17T^{2} \) |
| 19 | \( 1 + 4.99iT - 19T^{2} \) |
| 23 | \( 1 - 1.10iT - 23T^{2} \) |
| 29 | \( 1 + 1.96iT - 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 5.99T + 43T^{2} \) |
| 47 | \( 1 - 8.05iT - 47T^{2} \) |
| 53 | \( 1 + 1.97T + 53T^{2} \) |
| 59 | \( 1 - 1.44iT - 59T^{2} \) |
| 61 | \( 1 - 8.71iT - 61T^{2} \) |
| 67 | \( 1 - 3.92T + 67T^{2} \) |
| 71 | \( 1 + 4.55T + 71T^{2} \) |
| 73 | \( 1 + 7.55iT - 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 - 8.26T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−9.101304744803129054051659083992, −8.330249639394402300412170187430, −7.61572194819950549265180057283, −6.68122626469747173639442611635, −6.05151993606923606190083182475, −4.98117075158384369076160157358, −3.73604700327661269757209242525, −3.15511679872184560435257592612, −2.34169601284022830775207065383, −1.19162965239034999087776471513,
1.55485585527464917123489576599, 2.06081232696405164645956039040, 3.27385924713212113068501699207, 3.95732424938040324458426563304, 5.08210720929934039886211968777, 5.98045571918276531914261603086, 6.69722921127567004231705607876, 7.88348779532970034352876215151, 8.465131661156485385182384955157, 8.874167481277641829317916022470
