| L(s) = 1 | + i·3-s − 2i·5-s + 7-s − 9-s + 6i·13-s + 2·15-s − 2·17-s − 4i·19-s + i·21-s − 4·23-s + 25-s − i·27-s − 10i·29-s + 8·31-s − 2i·35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.894i·5-s + 0.377·7-s − 0.333·9-s + 1.66i·13-s + 0.516·15-s − 0.485·17-s − 0.917i·19-s + 0.218i·21-s − 0.834·23-s + 0.200·25-s − 0.192i·27-s − 1.85i·29-s + 1.43·31-s − 0.338i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.694554114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.694554114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.215735335379379103671220428565, −7.49143038830493183463750356596, −6.51308361275404631690947883969, −5.96007335712832069188690337281, −4.84631101945086933743168470072, −4.50976349643955564460695762722, −3.93585234750388257822869608901, −2.60072437451123026833339294951, −1.80898102363758655676791420763, −0.51952603050619733549010079545,
0.966433392124085248447454948145, 2.03712889503776471748621867268, 3.00009693610116623574686008236, 3.47877529801730529661271223687, 4.74899411191433050641089228391, 5.44025276299990909053513031805, 6.31271659530698073193551938332, 6.74931624508696532361323359352, 7.64228857986017205027178762763, 8.158964393260826818419465510316
