| L(s) = 1 | + 1.83·3-s + 1.31i·7-s + 0.368·9-s + 6.60i·11-s − 2.96·13-s − 2.69i·17-s + 4.97i·19-s + 2.41i·21-s − 3.73i·23-s − 4.83·27-s − 5.52i·29-s − 8.36·31-s + 12.1i·33-s − 7.19·37-s − 5.44·39-s + ⋯ |
| L(s) = 1 | + 1.05·3-s + 0.497i·7-s + 0.122·9-s + 1.99i·11-s − 0.822·13-s − 0.653i·17-s + 1.14i·19-s + 0.526i·21-s − 0.779i·23-s − 0.929·27-s − 1.02i·29-s − 1.50·31-s + 2.10i·33-s − 1.18·37-s − 0.871·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059555689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059555689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.83T + 3T^{2} \) |
| 7 | \( 1 - 1.31iT - 7T^{2} \) |
| 11 | \( 1 - 6.60iT - 11T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 2.69iT - 17T^{2} \) |
| 19 | \( 1 - 4.97iT - 19T^{2} \) |
| 23 | \( 1 + 3.73iT - 23T^{2} \) |
| 29 | \( 1 + 5.52iT - 29T^{2} \) |
| 31 | \( 1 + 8.36T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 + 3.77T + 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 - 8.77iT - 47T^{2} \) |
| 53 | \( 1 - 0.0464T + 53T^{2} \) |
| 59 | \( 1 + 1.02iT - 59T^{2} \) |
| 61 | \( 1 + 5.23iT - 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 9.35T + 71T^{2} \) |
| 73 | \( 1 + 12.4iT - 73T^{2} \) |
| 79 | \( 1 - 1.43T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 - 1.00iT - 97T^{2} \) |
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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
−8.988020713727945783066177076174, −7.82104731018427409906300698164, −7.60984625798170381980020174938, −6.78145830510937831189843846255, −5.76123090026713373513833624323, −4.90502016626414341744351006627, −4.20037869614001222795844609501, −3.23053932305011817361526909697, −2.27949658256324284503006077260, −1.88332490497894619239394099518,
0.23075089552270524924905819615, 1.61168736959472412720006747829, 2.75440784689415094341436710063, 3.37530040623848461754143949642, 3.98104920087643103623861789145, 5.26007307465514856037528404791, 5.76265952973458916114202412483, 6.92566664334379138470708631949, 7.42084083590489407245530206057, 8.303924271145831696559357579450
