L(s) = 1 | − 2.29·5-s − i·7-s + 4.02i·11-s + 1.82i·13-s − 0.430i·17-s − 5.01·19-s − 3.49·23-s + 0.274·25-s − 2.16·29-s − 2.10i·31-s + 2.29i·35-s + 2.19i·37-s − 4.35i·41-s + 12.0·43-s + 1.72·47-s + ⋯ |
L(s) = 1 | − 1.02·5-s − 0.377i·7-s + 1.21i·11-s + 0.506i·13-s − 0.104i·17-s − 1.14·19-s − 0.728·23-s + 0.0548·25-s − 0.401·29-s − 0.378i·31-s + 0.388i·35-s + 0.361i·37-s − 0.679i·41-s + 1.83·43-s + 0.251·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7640916969\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7640916969\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
good | 5 | \( 1 + 2.29T + 5T^{2} \) |
| 11 | \( 1 - 4.02iT - 11T^{2} \) |
| 13 | \( 1 - 1.82iT - 13T^{2} \) |
| 17 | \( 1 + 0.430iT - 17T^{2} \) |
| 19 | \( 1 + 5.01T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 + 2.16T + 29T^{2} \) |
| 31 | \( 1 + 2.10iT - 31T^{2} \) |
| 37 | \( 1 - 2.19iT - 37T^{2} \) |
| 41 | \( 1 + 4.35iT - 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 1.72T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 10.6iT - 59T^{2} \) |
| 61 | \( 1 + 8.44iT - 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 8.95T + 71T^{2} \) |
| 73 | \( 1 + 7.16T + 73T^{2} \) |
| 79 | \( 1 + 15.2iT - 79T^{2} \) |
| 83 | \( 1 - 13.4iT - 83T^{2} \) |
| 89 | \( 1 - 7.44iT - 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−7.84452241104305806483411267524, −7.31325070694768787387446761740, −6.68811750784413579081451618398, −5.87724719087858201989476311664, −4.81071975223443026328980501479, −4.16626164186867424774803988853, −3.81501610257247770517376195361, −2.53372303498945217355341310353, −1.71608066500172106406724688198, −0.27414636511902250888985365844,
0.73839132667634805591211698594, 2.12153050809292129741963040785, 3.08007876459752611852516958290, 3.79634774663639485330542497279, 4.42625108751074907785927222025, 5.46293343419416411762843562502, 6.04346611657383650056612398781, 6.76047101804973969573536146429, 7.79434213675650140803584084908, 8.087659890310824113437375965004