| L(s) = 1 | − 468. i·5-s − 81.2·7-s − 3.39e3i·11-s + 1.44e4i·13-s − 5.17e3·17-s − 4.60e4i·19-s − 6.75e4·23-s − 1.41e5·25-s − 2.53e4i·29-s − 1.23e5·31-s + 3.80e4i·35-s + 1.04e5i·37-s + 1.60e5·41-s − 3.78e5i·43-s + 6.83e5·47-s + ⋯ |
| L(s) = 1 | − 1.67i·5-s − 0.0895·7-s − 0.768i·11-s + 1.82i·13-s − 0.255·17-s − 1.54i·19-s − 1.15·23-s − 1.80·25-s − 0.193i·29-s − 0.746·31-s + 0.150i·35-s + 0.339i·37-s + 0.364·41-s − 0.726i·43-s + 0.960·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1796172184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1796172184\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 468. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 81.2T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.39e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.44e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 5.17e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.60e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 6.75e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.53e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.23e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.04e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 1.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.78e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 6.83e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.74e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.48e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 4.44e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.48e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.10e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.80e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.91e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 6.35e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.68e6T + 8.07e13T^{2} \) |
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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
−11.06830461476006334903904531170, −9.513523448632883709036566964471, −9.024081942739025581505318274509, −8.259924021859914204288397162294, −6.94174667125654435591313004274, −5.80188537949530809565561385401, −4.72046844386628123023177668068, −3.98814145762039143536520528650, −2.19071970795732074281682290380, −1.05051357886134919932336888944,
0.04229172022577073616344379124, 1.90516453018188568320736048158, 2.99835529590099847010075649238, 3.86921692275014075768440719179, 5.54297001215618363641816317454, 6.38153168676250612669467941752, 7.47048454886277958226755490300, 8.068832764716124302400775701645, 9.767942269781368880075269828884, 10.32261230545085091891757679430
