| L(s) = 1 | − 1.73i·3-s − 0.480i·5-s − 2.99·9-s + 5.76·11-s + 1.41i·13-s − 0.832·15-s − 23.0i·17-s − 10.6i·19-s − 6.59·23-s + 24.7·25-s + 5.19i·27-s − 6.20·29-s − 41.9i·31-s − 9.98i·33-s − 60.0·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.0961i·5-s − 0.333·9-s + 0.523·11-s + 0.109i·13-s − 0.0555·15-s − 1.35i·17-s − 0.561i·19-s − 0.286·23-s + 0.990·25-s + 0.192i·27-s − 0.213·29-s − 1.35i·31-s − 0.302i·33-s − 1.62·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.418106865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418106865\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.480iT - 25T^{2} \) |
| 11 | \( 1 - 5.76T + 121T^{2} \) |
| 13 | \( 1 - 1.41iT - 169T^{2} \) |
| 17 | \( 1 + 23.0iT - 289T^{2} \) |
| 19 | \( 1 + 10.6iT - 361T^{2} \) |
| 23 | \( 1 + 6.59T + 529T^{2} \) |
| 29 | \( 1 + 6.20T + 841T^{2} \) |
| 31 | \( 1 + 41.9iT - 961T^{2} \) |
| 37 | \( 1 + 60.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 48.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 19.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 82.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 92.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.99iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 2.20T + 4.48e3T^{2} \) |
| 71 | \( 1 + 80.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 13.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 64.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31.7iT - 9.40e3T^{2} \) |
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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
−10.18190012183553832911117027445, −9.203602696218617132730843579167, −8.532128050177480666089862387725, −7.33607095549464287871098575927, −6.80185922816732413610031786724, −5.64329515768145070546905316761, −4.64273812776444360566769021478, −3.30126735236535455547484100997, −2.04720873929289110101389776907, −0.54565602865809093558970573168,
1.55702724321282662859693209320, 3.18336062023681476811388620553, 4.08170910135754227809675784925, 5.19242707408416029659876238872, 6.19047641226297684640192485019, 7.12130268478400874399201519101, 8.409915623698066382594713309225, 8.877770209445028052789006307143, 10.19452322633573192650420725157, 10.45554383203990773515003721549
