| L(s) = 1 | + i·3-s + 2i·5-s − 7-s − 9-s + 2i·13-s − 2·15-s + 6·17-s − 4i·19-s − i·21-s − 4·23-s + 25-s − i·27-s − 6i·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 + 0.554i·13-s − 0.516·15-s + 1.45·17-s − 0.917i·19-s − 0.218i·21-s − 0.834·23-s + 0.200·25-s − 0.192i·27-s − 1.11i·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
| \(L(1)\) |
\(\approx\) |
\(1.922005174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922005174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 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
−8.279397665312073049335373723110, −7.40671623888464360597444339395, −6.95067797102855960568258859371, −5.94983842431006207501838394498, −5.60324904344828334472451245663, −4.35395701484166710581149497524, −3.89614060117994984325577992245, −2.88125332952870959049936553264, −2.35445385359391739769152291113, −0.72811319257996027822926713571,
0.831924491803121236226692459483, 1.47263088493981253453015684547, 2.77799802826630409595912644190, 3.46197395810664827940175595087, 4.51706386798544310890041215869, 5.25411710586797739099366113284, 5.98982745538633593043346979115, 6.52044462824515729705326468352, 7.65302120166007155431532517894, 7.985518312703168641727336835325
