| L(s) = 1 | + 3i·3-s − 16i·7-s − 9·9-s − 12·11-s + 38i·13-s + 126i·17-s + 20·19-s + 48·21-s − 168i·23-s − 27i·27-s − 30·29-s + 88·31-s − 36i·33-s − 254i·37-s − 114·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.863i·7-s − 0.333·9-s − 0.328·11-s + 0.810i·13-s + 1.79i·17-s + 0.241·19-s + 0.498·21-s − 1.52i·23-s − 0.192i·27-s − 0.192·29-s + 0.509·31-s − 0.189i·33-s − 1.12i·37-s − 0.468·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 16iT - 343T^{2} \) |
| 11 | \( 1 + 12T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 126iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 20T + 6.85e3T^{2} \) |
| 23 | \( 1 + 168iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 30T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 42T + 6.89e4T^{2} \) |
| 43 | \( 1 - 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 96iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 198iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 660T + 2.05e5T^{2} \) |
| 61 | \( 1 + 538T + 2.26e5T^{2} \) |
| 67 | \( 1 - 884iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 792T + 3.57e5T^{2} \) |
| 73 | \( 1 - 218iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 520T + 4.93e5T^{2} \) |
| 83 | \( 1 - 492iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 810T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3iT - 9.12e5T^{2} \) |
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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
−8.997227908618806238698417464403, −8.320512619116189334442244954610, −7.39831155439833277509557199522, −6.47860785121528150119841486973, −5.66644457398097043909352647892, −4.36044902760691209996649849335, −4.06816630419670133871129090520, −2.76798872083142334398790891413, −1.46409152055173266180041796203, 0,
1.30335450165696946029486186839, 2.59921033181442529508699353588, 3.23783747967452319517203150092, 4.87036306743413755010910335647, 5.49670706221724838979324756826, 6.35634617948902402157958787459, 7.44996286316676821297516429161, 7.88305431699451384043521813702, 8.982967439439460574362266857953
