| L(s) = 1 | + i·2-s − 4-s + 2i·5-s + 2i·7-s − i·8-s − 2·10-s + (−3 + 2i)13-s − 2·14-s + 16-s + 2·17-s + 6i·19-s − 2i·20-s − 4·23-s + 25-s + (−2 − 3i)26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.894i·5-s + 0.755i·7-s − 0.353i·8-s − 0.632·10-s + (−0.832 + 0.554i)13-s − 0.534·14-s + 0.250·16-s + 0.485·17-s + 1.37i·19-s − 0.447i·20-s − 0.834·23-s + 0.200·25-s + (−0.392 − 0.588i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.507552 + 0.948370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.507552 + 0.948370i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−12.38284341145745979941739153293, −11.74274292373543036929049707771, −10.34969921396913105751686373402, −9.634127354878424262414246398205, −8.401786004937940332109564926983, −7.48072179151526523019388320071, −6.42094053815453976712836435437, −5.54880825876954515067440994214, −4.07498965048745238112658744397, −2.49805887825902958986511875731,
0.942953026571404839167082239747, 2.88002558478575492595584477474, 4.42018691647936959828189349714, 5.20118278645230288588682717410, 6.88883468919043707757735434290, 8.100307463457145770790697528030, 9.021502634551145273383072217145, 10.08091823126173197286688880056, 10.76402192936562144044738310590, 12.10378650439117253917267311756
