| L(s) = 1 | + 3i·3-s − 5i·7-s − 9·9-s − 14·11-s − i·13-s + 46i·17-s + 19·19-s + 15·21-s − 46i·23-s − 27i·27-s − 14·29-s − 133·31-s − 42i·33-s + 258i·37-s + 3·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.269i·7-s − 0.333·9-s − 0.383·11-s − 0.0213i·13-s + 0.656i·17-s + 0.229·19-s + 0.155·21-s − 0.417i·23-s − 0.192i·27-s − 0.0896·29-s − 0.770·31-s − 0.221i·33-s + 1.14i·37-s + 0.0123·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.245673852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245673852\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5iT - 343T^{2} \) |
| 11 | \( 1 + 14T + 1.33e3T^{2} \) |
| 13 | \( 1 + iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 46iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 19T + 6.85e3T^{2} \) |
| 23 | \( 1 + 46iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 14T + 2.43e4T^{2} \) |
| 31 | \( 1 + 133T + 2.97e4T^{2} \) |
| 37 | \( 1 - 258iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 84T + 6.89e4T^{2} \) |
| 43 | \( 1 + 167iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 410iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 456iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 194T + 2.05e5T^{2} \) |
| 61 | \( 1 + 17T + 2.26e5T^{2} \) |
| 67 | \( 1 + 653iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 828T + 3.57e5T^{2} \) |
| 73 | \( 1 + 570iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 552T + 4.93e5T^{2} \) |
| 83 | \( 1 - 142iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 841iT - 9.12e5T^{2} \) |
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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
−9.236608865731083763355389776747, −8.490078953783317211010443185000, −7.66770248606964398152314325281, −6.72043598227828869950645212363, −5.75688026059346262893877801932, −4.92064878766796327484429902369, −3.98829699584281758309677404845, −3.10458170886009886005081542092, −1.84938106257661067139180973486, −0.33226466449880231217591547039,
0.997453533672714072584271601255, 2.22803927767488163653189640446, 3.13964558403526928726489327418, 4.37812333010377834639422300518, 5.46025372065920046688909614585, 6.09493377238305412731079992222, 7.28703502884847903674895974005, 7.63425352151050579720158848178, 8.777134374669988223617351143887, 9.345351384744648606772442839705
