| L(s) = 1 | + i·3-s − i·5-s + 4·7-s − 9-s + 6i·13-s + 15-s − 2·17-s + 4i·19-s + 4i·21-s − 8·23-s − 25-s − i·27-s + 6i·29-s − 4i·35-s − 6i·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s + 1.51·7-s − 0.333·9-s + 1.66i·13-s + 0.258·15-s − 0.485·17-s + 0.917i·19-s + 0.872i·21-s − 1.66·23-s − 0.200·25-s − 0.192i·27-s + 1.11i·29-s − 0.676i·35-s − 0.986i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.398328585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398328585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.631708038465271509636149152327, −8.254725105928943268422837281103, −7.42998395045456739396150145840, −6.49170980420905646347835011813, −5.64187712399003463256000855771, −4.80718631124847877747972853011, −4.35416871791474225160972864503, −3.60331562053031336699909715989, −2.04407006503562681316775408195, −1.57224830848052748165244866993,
0.37767978696971425867334090635, 1.68427033679691729673500219140, 2.44424211886591476756172329866, 3.43008231842520856884765082610, 4.52925597230107979277564716679, 5.24683298351127885171103339580, 5.97813839942630910510714627822, 6.82654893591999681737315619370, 7.60063200755130718247577608228, 8.300104031527466841121200382345
