L(s) = 1 | + 2.73i·5-s + 7-s + 6.19i·11-s + 4.73·17-s + 0.535i·19-s + 5.26·23-s − 2.46·25-s − 3.46i·29-s + 8.92·31-s + 2.73i·35-s + 10i·37-s − 4.73·41-s − 12.9i·43-s + 6.92·47-s + 49-s + ⋯ |
L(s) = 1 | + 1.22i·5-s + 0.377·7-s + 1.86i·11-s + 1.14·17-s + 0.122i·19-s + 1.09·23-s − 0.492·25-s − 0.643i·29-s + 1.60·31-s + 0.461i·35-s + 1.64i·37-s − 0.739·41-s − 1.97i·43-s + 1.01·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.120556649\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.120556649\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 6.19iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 - 0.535iT - 19T^{2} \) |
| 23 | \( 1 - 5.26T + 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 + 12.9iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 2.92iT - 59T^{2} \) |
| 61 | \( 1 - 5.46iT - 61T^{2} \) |
| 67 | \( 1 + 0.535iT - 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 - 12.3iT - 83T^{2} \) |
| 89 | \( 1 - 4.73T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.544238164624479043183743753189, −7.76903900194751734669364283733, −7.06574568360109192412322550533, −6.76480758391907683435642350639, −5.69837162815793733743422769822, −4.86226879844422622990943654673, −4.12431909850271278332690868730, −3.06297750811738445298997105722, −2.39894017068598488140019256689, −1.30734350205996572313445701465,
0.71466605925668100200645110581, 1.28920741656391630514287370364, 2.80813150945835635471898823782, 3.55624608294188957560913060744, 4.58567366077884012270527182137, 5.27241543775451807756315950856, 5.81376226581405424329276779281, 6.68688072455456636039734815255, 7.81586541594809919423000842980, 8.236585619146675614517323225458