L(s) = 1 | + 3-s − 5-s − 2.20i·7-s + 9-s + 4.04i·11-s + 1.23i·13-s − 15-s − 4.40·17-s + (3.61 + 2.44i)19-s − 2.20i·21-s − 9.21i·23-s + 25-s + 27-s − 8.22i·29-s − 7.30·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.833i·7-s + 0.333·9-s + 1.22i·11-s + 0.342i·13-s − 0.258·15-s − 1.06·17-s + (0.828 + 0.559i)19-s − 0.481i·21-s − 1.92i·23-s + 0.200·25-s + 0.192·27-s − 1.52i·29-s − 1.31·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0705 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0705 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.506719843\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.506719843\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-3.61 - 2.44i)T \) |
good | 7 | \( 1 + 2.20iT - 7T^{2} \) |
| 11 | \( 1 - 4.04iT - 11T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 23 | \( 1 + 9.21iT - 23T^{2} \) |
| 29 | \( 1 + 8.22iT - 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 + 3.55iT - 37T^{2} \) |
| 41 | \( 1 + 4.31iT - 41T^{2} \) |
| 43 | \( 1 - 7.01iT - 43T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 - 7.55iT - 53T^{2} \) |
| 59 | \( 1 + 0.470T + 59T^{2} \) |
| 61 | \( 1 - 0.970T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 0.320T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 9.32iT - 83T^{2} \) |
| 89 | \( 1 + 4.17iT - 89T^{2} \) |
| 97 | \( 1 + 11.6iT - 97T^{2} \) |
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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
−8.038473908354150361183074591385, −7.38860438563644264447480941104, −6.95619824928119637377299223908, −6.11479588857013366317994475188, −4.87503307003147236247681883048, −4.25672501704943455923747710064, −3.77651672560422188273761207484, −2.55729103044027037763700092959, −1.81874325317211796570306231403, −0.39872999844512265870289524249,
1.16909366558798210579904657097, 2.32767630914126471896556659350, 3.28985280153365182753700239651, 3.63180122063713064377270651279, 4.96908040913432170902691057684, 5.46028258840250214113111456211, 6.36145344893073512754548260596, 7.23295552679284233379082733435, 7.82844822018767924233697730284, 8.638276780693573574736023800406