L(s) = 1 | + 3.16·3-s + 3.16i·7-s + 7.00·9-s − 6·13-s + 2i·17-s + 6.32i·19-s + 10.0i·21-s − 3.16i·23-s + 12.6·27-s + 4i·29-s + 6.32·31-s − 2·37-s − 18.9·39-s + 3.16·43-s + 9.48i·47-s + ⋯ |
L(s) = 1 | + 1.82·3-s + 1.19i·7-s + 2.33·9-s − 1.66·13-s + 0.485i·17-s + 1.45i·19-s + 2.18i·21-s − 0.659i·23-s + 2.43·27-s + 0.742i·29-s + 1.13·31-s − 0.328·37-s − 3.03·39-s + 0.482·43-s + 1.38i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.364802751\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.364802751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 7 | \( 1 - 3.16iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 3.16iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 6.32iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.770891116126454484493002701173, −8.148034568426103553012042649357, −7.63207601846752260828474295460, −6.78037331110222363925088264491, −5.78085603840777725777650687748, −4.78821890117484729333675584205, −3.98567102474136674751072216954, −2.94230703221830713069578990197, −2.47333584113622307697706320404, −1.63140069083734976278693283776,
0.77837134173139944058701293854, 2.20361431406323904774540157763, 2.75867623278001903763015005684, 3.71151801559659925677413396085, 4.42731584103857270669272915412, 5.12877633661462465813860435656, 6.86092700984071856030101350589, 7.12322068151770369278093988828, 7.78287744359182428335517612907, 8.449730418675149788457395049445