| L(s) = 1 | − 5-s − 3.16i·7-s − 1.16i·11-s + 5.88i·13-s + 1.64i·17-s − 1.64·19-s + 25-s − 8.32·29-s + 4.32i·31-s + 3.16i·35-s + 2.59i·37-s − 7.53i·41-s + 8.48·43-s − 10.1·47-s − 3.00·49-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.19i·7-s − 0.350i·11-s + 1.63i·13-s + 0.398i·17-s − 0.377·19-s + 0.200·25-s − 1.54·29-s + 0.776i·31-s + 0.534i·35-s + 0.427i·37-s − 1.17i·41-s + 1.29·43-s − 1.47·47-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.021357748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.021357748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 3.16iT - 7T^{2} \) |
| 11 | \( 1 + 1.16iT - 11T^{2} \) |
| 13 | \( 1 - 5.88iT - 13T^{2} \) |
| 17 | \( 1 - 1.64iT - 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 - 4.32iT - 31T^{2} \) |
| 37 | \( 1 - 2.59iT - 37T^{2} \) |
| 41 | \( 1 + 7.53iT - 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 2.32T + 53T^{2} \) |
| 59 | \( 1 - 13.1iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 0.324iT - 79T^{2} \) |
| 83 | \( 1 + 2.32iT - 83T^{2} \) |
| 89 | \( 1 - 16.0iT - 89T^{2} \) |
| 97 | \( 1 + 0.324T + 97T^{2} \) |
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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
−8.935169753856172378223512254162, −8.173708067209041811342001535665, −7.26757090213357632822846747908, −6.89768323054707182335242201879, −5.99199820847169731961597248630, −4.90949326170931489620175254656, −4.00403002887299148743589542691, −3.69810629251268708048125432013, −2.23406789782428952933086228177, −1.11375305170958920783474797520,
0.35687270948545598255833231105, 1.98944776833456399006603302557, 2.87775203100213665603601178107, 3.71494411901149373756330165485, 4.85229688779682700590153100428, 5.52175212415052444395198024694, 6.17714526765027905119971615209, 7.21864569136816423772494069762, 7.998991933560847394995985859162, 8.418829734308882699436584717287
