| L(s) = 1 | + 3-s + 1.73i·5-s + 9-s − 5.19i·11-s + 6.92i·13-s + 1.73i·15-s + 3.46i·17-s − 2·19-s + 6.92i·23-s + 2.00·25-s + 27-s − 9·29-s − 31-s − 5.19i·33-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.774i·5-s + 0.333·9-s − 1.56i·11-s + 1.92i·13-s + 0.447i·15-s + 0.840i·17-s − 0.458·19-s + 1.44i·23-s + 0.400·25-s + 0.192·27-s − 1.67·29-s − 0.179·31-s − 0.904i·33-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.773832851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773832851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 - 6.92iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 6.92iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 1.73iT - 79T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−9.021969942521490074614525745408, −8.622971019739054333990298511950, −7.61447029283636806153977649183, −6.92115446833376555040757999752, −6.20633624920199868461734217867, −5.38803813851764505679459487226, −3.95388238717926402880942663635, −3.63588145971488819911007379759, −2.48811849411808031473204208142, −1.51893147003307601688768426475,
0.54503241457184963027589456546, 1.94137547141932841602340435805, 2.83232988881421012280967612273, 3.92849097981636598163748821257, 4.85689434590501233970516167507, 5.34634570947769757537990730811, 6.56984864773079163300538120091, 7.45291502206463489730595841104, 7.962262603764789126290965356304, 8.819208174125599229354587656741
