| L(s) = 1 | + 0.710·5-s + 3.12i·7-s − 16.6i·11-s + 18.3·13-s + 9.69·17-s − 8.20i·19-s + 2.24i·23-s − 24.4·25-s − 41.6·29-s + 24.9i·31-s + 2.21i·35-s − 40.3·37-s + 51.3·41-s − 65.4i·43-s − 33.8i·47-s + ⋯ |
| L(s) = 1 | + 0.142·5-s + 0.446i·7-s − 1.50i·11-s + 1.41·13-s + 0.570·17-s − 0.431i·19-s + 0.0978i·23-s − 0.979·25-s − 1.43·29-s + 0.805i·31-s + 0.0634i·35-s − 1.09·37-s + 1.25·41-s − 1.52i·43-s − 0.719i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.916250939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.916250939\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.710T + 25T^{2} \) |
| 7 | \( 1 - 3.12iT - 49T^{2} \) |
| 11 | \( 1 + 16.6iT - 121T^{2} \) |
| 13 | \( 1 - 18.3T + 169T^{2} \) |
| 17 | \( 1 - 9.69T + 289T^{2} \) |
| 19 | \( 1 + 8.20iT - 361T^{2} \) |
| 23 | \( 1 - 2.24iT - 529T^{2} \) |
| 29 | \( 1 + 41.6T + 841T^{2} \) |
| 31 | \( 1 - 24.9iT - 961T^{2} \) |
| 37 | \( 1 + 40.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 51.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 65.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 33.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 90.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 76.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 2.71T + 3.72e3T^{2} \) |
| 67 | \( 1 - 39.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 38.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 131. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 38.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 24.3T + 9.40e3T^{2} \) |
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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
−9.021046646726000806140547390104, −8.729862079635466802760670822424, −7.82240390613119145753594324335, −6.78578677601985879470288602382, −5.72127229075802002804644119116, −5.53554921613787954844873042231, −3.87928624268030439438380709232, −3.29522077104879942926147822549, −1.93165685315087173109146003648, −0.61591712694013038365836341885,
1.19020544969034686347134959408, 2.22969047219060243897357917481, 3.70084943311142375437557490887, 4.27782314125250836313609128597, 5.52095838249317815193100339605, 6.21987687596566721151647197121, 7.32648225247660957084994941534, 7.78245146896862380784180440115, 8.885066077046156761969483842781, 9.665921215806563965009115546674
