| L(s) = 1 | + 1.35i·3-s + 0.648i·7-s + 1.17·9-s + 3.35·11-s + 4.17i·13-s − 4.82i·17-s − 6.82·19-s − 0.876·21-s + 5.52i·23-s + 5.64i·27-s + 29-s − 2.82·31-s + 4.53i·33-s + 10.2i·37-s − 5.64·39-s + ⋯ |
| L(s) = 1 | + 0.780i·3-s + 0.244i·7-s + 0.390·9-s + 1.01·11-s + 1.15i·13-s − 1.16i·17-s − 1.56·19-s − 0.191·21-s + 1.15i·23-s + 1.08i·27-s + 0.185·29-s − 0.506·31-s + 0.788i·33-s + 1.68i·37-s − 0.903·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743306986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743306986\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.35iT - 3T^{2} \) |
| 7 | \( 1 - 0.648iT - 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 - 4.17iT - 13T^{2} \) |
| 17 | \( 1 + 4.82iT - 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 5.52iT - 23T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 10.2iT - 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 + 5.69iT - 43T^{2} \) |
| 47 | \( 1 - 2.64iT - 47T^{2} \) |
| 53 | \( 1 + 2.87iT - 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 + 1.52iT - 67T^{2} \) |
| 71 | \( 1 + 8.87T + 71T^{2} \) |
| 73 | \( 1 - 9.69iT - 73T^{2} \) |
| 79 | \( 1 + 8.99T + 79T^{2} \) |
| 83 | \( 1 - 1.94iT - 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
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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
−9.063119354427458163875541908440, −8.581473287291675528289791392970, −7.29974177682906855388850052778, −6.82982004722803072495128277070, −5.95029195213226924727302777228, −4.95118582881574650965086419213, −4.24674496459150001812706178548, −3.70056172585261031554346655812, −2.43302752090254754718850329228, −1.35396691623909700858900142909,
0.58194129699924828455124599468, 1.66416224237692529743728135055, 2.57693226430316908732589183835, 3.94532081001897662350759861983, 4.33055597049063262116611691155, 5.74895585758420298926867317282, 6.28492155182959352457335551343, 7.00894716050268245326537207728, 7.72371142317507397484942783856, 8.468520773843406042105779271291
