| L(s) = 1 | + (−2.5 + 1.65i)3-s − 3.31·5-s − 7i·7-s + (3.5 − 8.29i)9-s + 19.8·11-s − 13i·13-s + (8.29 − 5.5i)15-s + 23.2i·17-s − 30i·19-s + (11.6 + 17.5i)21-s − 26.5i·23-s − 14·25-s + (4.99 + 26.5i)27-s − 33.1i·29-s − 10i·31-s + ⋯ |
| L(s) = 1 | + (−0.833 + 0.552i)3-s − 0.663·5-s − i·7-s + (0.388 − 0.921i)9-s + 1.80·11-s − i·13-s + (0.552 − 0.366i)15-s + 1.36i·17-s − 1.57i·19-s + (0.552 + 0.833i)21-s − 1.15i·23-s − 0.560·25-s + (0.185 + 0.982i)27-s − 1.14i·29-s − 0.322i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.816247 - 0.438059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.816247 - 0.438059i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.5 - 1.65i)T \) |
| 13 | \( 1 + 13iT \) |
| good | 5 | \( 1 + 3.31T + 25T^{2} \) |
| 7 | \( 1 + 7iT - 49T^{2} \) |
| 11 | \( 1 - 19.8T + 121T^{2} \) |
| 17 | \( 1 - 23.2iT - 289T^{2} \) |
| 19 | \( 1 + 30iT - 361T^{2} \) |
| 23 | \( 1 + 26.5iT - 529T^{2} \) |
| 29 | \( 1 + 33.1iT - 841T^{2} \) |
| 31 | \( 1 + 10iT - 961T^{2} \) |
| 37 | \( 1 - 13iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 55T + 1.84e3T^{2} \) |
| 47 | \( 1 + 49.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 39.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 92.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 14T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 29.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 54T + 6.24e3T^{2} \) |
| 83 | \( 1 + 33.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 19.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 78iT - 9.40e3T^{2} \) |
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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
−12.32477437347409773871271465325, −11.43853223898962124179662068435, −10.67950581117801860292118147659, −9.701816238400614464165710109525, −8.430304474244736172697694380527, −7.03564369236225245274486602715, −6.13281339335208586357576557425, −4.45241326931927976958809430496, −3.76215809825969821746871058000, −0.71432651936448818814285899130,
1.63235638870685096934679585974, 3.84842843465239551134682595147, 5.32411919136326110299897242464, 6.46979149565321526394305327137, 7.39132493555182713693235329213, 8.752085376162274458565104735048, 9.741632869855025924077213522518, 11.49323875960718543780209976946, 11.72497599533462783189665929009, 12.42460542018230087651356878238
