L(s) = 1 | + 3-s + 5-s + 9-s − 3.46i·11-s + 15-s + (−4 − 1.73i)19-s + 3.46i·23-s + 25-s + 27-s − 6.92i·29-s − 2·31-s − 3.46i·33-s − 6.92i·37-s + 3.46i·41-s − 10.3i·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.333·9-s − 1.04i·11-s + 0.258·15-s + (−0.917 − 0.397i)19-s + 0.722i·23-s + 0.200·25-s + 0.192·27-s − 1.28i·29-s − 0.359·31-s − 0.603i·33-s − 1.13i·37-s + 0.541i·41-s − 1.58i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.163613643\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.163613643\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (4 + 1.73i)T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 10.3iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−8.245036317676522673683277458715, −7.51388952513866013267515263990, −6.72575109735797958936065707001, −5.93047201228314773852540650754, −5.34552681182333840847444051958, −4.23620559613164206393440062539, −3.58311703895617436239772349244, −2.61610068872793840162000454166, −1.87193557791918760841205381417, −0.53159866053781738224170611867,
1.33240923041998345813823778269, 2.18985871907087045381540913148, 2.97477243240723019796176203917, 4.03525711104170151474183718575, 4.68546231262043105634160233819, 5.51576277471748180185955958968, 6.52634394994874698322262885455, 6.95926625021691650094962106942, 7.87787115486009899093262521863, 8.476774708166337686592410230821