| L(s) = 1 | − 23.1·5-s − 4.13·7-s + 212.·11-s − 36.7i·13-s + 99.6i·17-s + 316. i·19-s − 878. i·23-s − 89.8·25-s − 596.·29-s + 1.13e3·31-s + 95.5·35-s − 998. i·37-s + 2.32e3i·41-s + 181. i·43-s + 2.48e3i·47-s + ⋯ |
| L(s) = 1 | − 0.925·5-s − 0.0843·7-s + 1.75·11-s − 0.217i·13-s + 0.344i·17-s + 0.877i·19-s − 1.66i·23-s − 0.143·25-s − 0.709·29-s + 1.18·31-s + 0.0780·35-s − 0.729i·37-s + 1.38i·41-s + 0.0983i·43-s + 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.785506452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785506452\) |
| \(L(3)\) |
|
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 | \( 1 + 23.1T + 625T^{2} \) |
| 7 | \( 1 + 4.13T + 2.40e3T^{2} \) |
| 11 | \( 1 - 212.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 36.7iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 99.6iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 316. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 878. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 596.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.13e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 998. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.32e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 181. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.48e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.46e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 2.33e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 3.53e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.23e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.78e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.25e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.51e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 3.01e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 7.16e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 852.T + 8.85e7T^{2} \) |
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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
−9.209370523096326011679438331743, −8.286170599144954280729149416936, −7.74698675409934271376475813035, −6.56484067983997765622190899674, −6.15023219040627534039020784312, −4.65277627736412292212446914986, −4.01115224078992189069212971370, −3.17576242272507967919561437371, −1.71885046276388009916767777375, −0.60222555070223973365082842234,
0.66696650612392348295631863352, 1.78304032078404364600572356089, 3.28774332089335833797896292622, 3.93870721340864102324763325907, 4.82374159870319546321658201921, 6.00263528395018842764467609351, 6.91917402049871163979914808895, 7.48929370172045701243114085611, 8.534558474972468983439829184623, 9.226948467520900779672403161146
