L(s) = 1 | + 3.20i·3-s − 7.26·9-s + 4.20·11-s − 0.204i·13-s − 5.06i·17-s + 1.06·19-s + 2.14i·23-s − 13.6i·27-s + 7.47·29-s + 8.47·31-s + 13.4i·33-s + 10.6i·37-s + 0.654·39-s − 10.5·41-s + 8.26i·43-s + ⋯ |
L(s) = 1 | + 1.85i·3-s − 2.42·9-s + 1.26·11-s − 0.0566i·13-s − 1.22i·17-s + 0.244·19-s + 0.446i·23-s − 2.63i·27-s + 1.38·29-s + 1.52·31-s + 2.34i·33-s + 1.74i·37-s + 0.104·39-s − 1.64·41-s + 1.26i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.803181399\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.803181399\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3.20iT - 3T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 + 0.204iT - 13T^{2} \) |
| 17 | \( 1 + 5.06iT - 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 2.14iT - 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 8.26iT - 43T^{2} \) |
| 47 | \( 1 - 3.26iT - 47T^{2} \) |
| 53 | \( 1 - 5.67iT - 53T^{2} \) |
| 59 | \( 1 - 1.20T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 0.591T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 + 3.88iT - 83T^{2} \) |
| 89 | \( 1 + 9.26T + 89T^{2} \) |
| 97 | \( 1 - 1.33iT - 97T^{2} \) |
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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
−8.628158360211055376595108480644, −8.254835874270244581705573260499, −6.97921214920020553935639347472, −6.29669448533828129886654648258, −5.45294980415079751788883952834, −4.55481878853207117730099593427, −4.41829187377728131514946228024, −3.20287343055486685325555906391, −2.88611469233271075219875779478, −1.12769976906441931354383696851,
0.55491095934999011335354552096, 1.48558879067311847363120364859, 2.16345310353384900381509091398, 3.19132935650230238759468903954, 4.10479228534858539455911841315, 5.26078108304876190952122123159, 6.15631227580328553184739213045, 6.58033720023524029728297062040, 7.05905322000430598290366646893, 7.981614779041021617258623056616