| L(s) = 1 | + (−1.70 − 0.292i)3-s − 0.585i·5-s + i·7-s + (2.82 + i)9-s − 2·11-s − 0.585·13-s + (−0.171 + i)15-s − 0.828i·17-s − 2.24i·19-s + (0.292 − 1.70i)21-s − 4·23-s + 4.65·25-s + (−4.53 − 2.53i)27-s + 0.828i·29-s + 6.82i·31-s + ⋯ |
| L(s) = 1 | + (−0.985 − 0.169i)3-s − 0.261i·5-s + 0.377i·7-s + (0.942 + 0.333i)9-s − 0.603·11-s − 0.162·13-s + (−0.0442 + 0.258i)15-s − 0.200i·17-s − 0.514i·19-s + (0.0639 − 0.372i)21-s − 0.834·23-s + 0.931·25-s + (−0.872 − 0.487i)27-s + 0.153i·29-s + 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 + 0.292i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 0.585iT - 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 + 2.24iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 0.828iT - 29T^{2} \) |
| 31 | \( 1 - 6.82iT - 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 6.48iT - 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 - 9.31iT - 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 - 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 8.58T + 83T^{2} \) |
| 89 | \( 1 + 3.65iT - 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−9.160577354132807901435745433663, −8.425340149454521764545455102857, −7.35132446693470184016777328012, −6.75905704268190864936343737085, −5.67487782465695139011411285817, −5.14734717659823155056595530341, −4.25116060557960901461337905659, −2.84783506185346804193196464432, −1.54237667964816931850509329725, 0,
1.59684067971655242224447331755, 3.10324100948562705644926905010, 4.23643958493199782952235130796, 4.99257115674037405051011188130, 5.97718110705282841912229645290, 6.59974374370759535671892653190, 7.56172030681361702655302401597, 8.245605509868299300298504834837, 9.586879195057445952912072852466
