L(s) = 1 | − 5-s − 3.16i·7-s − 5.16i·11-s + 3.05i·13-s + 7.30i·17-s + 7.30·19-s + 25-s + 4.32·29-s + 8.32i·31-s + 3.16i·35-s − 11.5i·37-s − 10.3i·41-s + 8.48·43-s − 1.18·47-s − 3.00·49-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.19i·7-s − 1.55i·11-s + 0.848i·13-s + 1.77i·17-s + 1.67·19-s + 0.200·25-s + 0.803·29-s + 1.49i·31-s + 0.534i·35-s − 1.89i·37-s − 1.61i·41-s + 1.29·43-s − 0.172·47-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.543748058\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.543748058\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + 3.16iT - 7T^{2} \) |
| 11 | \( 1 + 5.16iT - 11T^{2} \) |
| 13 | \( 1 - 3.05iT - 13T^{2} \) |
| 17 | \( 1 - 7.30iT - 17T^{2} \) |
| 19 | \( 1 - 7.30T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4.32T + 29T^{2} \) |
| 31 | \( 1 - 8.32iT - 31T^{2} \) |
| 37 | \( 1 + 11.5iT - 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 1.18T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6.83iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 12.3iT - 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 1.87iT - 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.630798485512463431910530183249, −7.76723572184700664923757151062, −7.22286326188322200583161368566, −6.33381411492440253711795950062, −5.61859042132559132323207378934, −4.53783652431009991920181676307, −3.70116068799211427084607979300, −3.25412659549503768301830785058, −1.61491148899301569721629135857, −0.58184783937199424772887929117,
1.12515719127505089303640734221, 2.62610731972336804167084421317, 2.97288406034477227594862936209, 4.45856500575840239529729435556, 5.01431351116814259464399527637, 5.76167172680446302020463668838, 6.77209114551827279231746062463, 7.58881894995609991306677266344, 7.977861141001479577653829033523, 9.101808003889764179924612124819