| L(s) = 1 | − 3-s − 2i·5-s + i·7-s − 2·9-s − i·11-s + 2i·15-s − 3·17-s + 7i·19-s − i·21-s + 23-s + 25-s + 5·27-s + 3·29-s + 8i·31-s + i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894i·5-s + 0.377i·7-s − 0.666·9-s − 0.301i·11-s + 0.516i·15-s − 0.727·17-s + 1.60i·19-s − 0.218i·21-s + 0.208·23-s + 0.200·25-s + 0.962·27-s + 0.557·29-s + 1.43i·31-s + 0.174i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.169483519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169483519\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 7iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 11iT - 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 9iT - 59T^{2} \) |
| 61 | \( 1 + 9T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + iT - 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
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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
−8.775857208941694997854218795872, −8.238588915009119756611839499688, −7.21161694358873745265204337167, −6.25884178590984143921231260338, −5.61678685939426403191630930048, −5.04639602240798195883222344416, −4.11152182723379111311316446364, −3.06382502924377567122370344832, −1.86499723891215150004652425731, −0.62257875241308956173529330689,
0.75473133753959104250392758937, 2.46185034718936909312431085688, 3.00876291063615817352521978514, 4.32886154077419110912420838471, 4.91748334996166143883068496785, 6.07852286350414379520734477397, 6.51475432654841634880373092971, 7.26919934834921942942779788910, 8.021034684646433357044593492149, 9.036735643273724545879808530933
