| L(s) = 1 | + 0.326i·3-s + (−0.278 + 2.21i)5-s + (−2.25 + 1.38i)7-s + 2.89·9-s + 4.91i·11-s − 4.89·13-s + (−0.724 − 0.0908i)15-s − 6.14·17-s + 4.72·19-s + (−0.452 − 0.735i)21-s + 6.30·23-s + (−4.84 − 1.23i)25-s + 1.92i·27-s − 4.66·29-s − 4.98·31-s + ⋯ |
| L(s) = 1 | + 0.188i·3-s + (−0.124 + 0.992i)5-s + (−0.852 + 0.523i)7-s + 0.964·9-s + 1.48i·11-s − 1.35·13-s + (−0.187 − 0.0234i)15-s − 1.49·17-s + 1.08·19-s + (−0.0986 − 0.160i)21-s + 1.31·23-s + (−0.969 − 0.247i)25-s + 0.370i·27-s − 0.866·29-s − 0.895·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6295257929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6295257929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.278 - 2.21i)T \) |
| 7 | \( 1 + (2.25 - 1.38i)T \) |
| good | 3 | \( 1 - 0.326iT - 3T^{2} \) |
| 11 | \( 1 - 4.91iT - 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 2.88iT - 37T^{2} \) |
| 41 | \( 1 + 7.38iT - 41T^{2} \) |
| 43 | \( 1 - 6.20T + 43T^{2} \) |
| 47 | \( 1 + 5.09iT - 47T^{2} \) |
| 53 | \( 1 + 1.63iT - 53T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 - 8.50iT - 61T^{2} \) |
| 67 | \( 1 + 3.67T + 67T^{2} \) |
| 71 | \( 1 + 1.27iT - 71T^{2} \) |
| 73 | \( 1 + 8.00T + 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 + 0.484iT - 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 + 6.14T + 97T^{2} \) |
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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
−9.437915452475794943176734791090, −9.108256124533098331295183956610, −7.44157747999740075061817525135, −7.20207116067921561367335813381, −6.69005608754610701227679310477, −5.46222749402905203057163734828, −4.64188272073998058073777797038, −3.76726117765620761713957963903, −2.69451191308818034722954109577, −1.99485476242296968746993184769,
0.22140844021431952706507927452, 1.26248644084452607459966398597, 2.69686554481226677424222570332, 3.72787475009871278376265784923, 4.55844503730826608472996299287, 5.35240320398270565558721749047, 6.29184654453732805835881042943, 7.21865892858456786353201473677, 7.62439599877543157884076637915, 8.782484162446953680864540942074
