| L(s) = 1 | − 3.41i·3-s − 0.828i·7-s − 8.65·9-s − 2·11-s − 6.24i·13-s − 0.828i·17-s + 19-s − 2.82·21-s − 6i·23-s + 19.3i·27-s + 6.48·29-s − 6.82·31-s + 6.82i·33-s + 1.75i·37-s − 21.3·39-s + ⋯ |
| L(s) = 1 | − 1.97i·3-s − 0.313i·7-s − 2.88·9-s − 0.603·11-s − 1.73i·13-s − 0.200i·17-s + 0.229·19-s − 0.617·21-s − 1.25i·23-s + 3.71i·27-s + 1.20·29-s − 1.22·31-s + 1.18i·33-s + 0.288i·37-s − 3.41·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8589734317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8589734317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.41iT - 3T^{2} \) |
| 7 | \( 1 + 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6.24iT - 13T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 1.75iT - 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 - 4.82iT - 43T^{2} \) |
| 47 | \( 1 - 4.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.07iT - 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 3.41iT - 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 2.48iT - 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 - 6.48T + 89T^{2} \) |
| 97 | \( 1 - 10.2iT - 97T^{2} \) |
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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
−8.337558176029440218748459778649, −7.73684991762181895402985938035, −7.34050816341826362903023592435, −6.30083966969604342660653094607, −5.80011625532092285750856883137, −4.82235230169950396525764833131, −3.08389291244899957566431123968, −2.60593289889015598548523966471, −1.26901289972120952857082027657, −0.32077540743513859221672103138,
2.15920137626435536748547054814, 3.29583035949237515835783238268, 4.02068940598534697596547161893, 4.81851264190098231838596317870, 5.47138925203447059176193359816, 6.31122210658978051780581285669, 7.51892635823834459125244391982, 8.581447609941493714769059520429, 9.150788408829538254508129693501, 9.625371532495666727369105405252
