L(s) = 1 | − 2.44i·2-s − i·3-s − 3.99·4-s − 2.44·6-s − i·7-s + 4.89i·8-s − 9-s + 2.44·11-s + 3.99i·12-s − 4.44i·13-s − 2.44·14-s + 3.99·16-s − 5.44i·17-s + 2.44i·18-s − 4.44·19-s + ⋯ |
L(s) = 1 | − 1.73i·2-s − 0.577i·3-s − 1.99·4-s − 0.999·6-s − 0.377i·7-s + 1.73i·8-s − 0.333·9-s + 0.738·11-s + 1.15i·12-s − 1.23i·13-s − 0.654·14-s + 0.999·16-s − 1.32i·17-s + 0.577i·18-s − 1.02·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8118485589\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8118485589\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 2 | \( 1 + 2.44iT - 2T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 4.44iT - 13T^{2} \) |
| 17 | \( 1 + 5.44iT - 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.101T + 31T^{2} \) |
| 37 | \( 1 - 3.89iT - 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 8.44iT - 47T^{2} \) |
| 53 | \( 1 + 0.550iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 0.651T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 - 5.34iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 15.2iT - 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 3.10iT - 97T^{2} \) |
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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
−9.114377350718979974128326451525, −8.007651587249542167796686229345, −7.27302232969621149920782624116, −6.14420670425830911227756678180, −5.11327464293658785146782019036, −4.14674997090807687427082283072, −3.28452339813390454673653613947, −2.43814817705597190601688730610, −1.36190589172820296312780653081, −0.31828742155593581477791918896,
1.99624348529175202437369518805, 3.92561768391038403023338709540, 4.21699984484430118904393884093, 5.35915690848169769851185601515, 6.08555754465443678207453273897, 6.63385146718359636215877187864, 7.52329470492662922288461477690, 8.408611473768957982706168125547, 9.057423086519731844749088856144, 9.397130858912827409389904862329