| L(s) = 1 | + 1.69·2-s + 0.875·4-s + 5-s − 1.30·7-s − 1.90·8-s + 1.69·10-s + 5.55·11-s − 4.25·13-s − 2.21·14-s − 4.98·16-s + 2.79·17-s − 0.360·19-s + 0.875·20-s + 9.42·22-s + 8.41·23-s + 25-s − 7.21·26-s − 1.14·28-s + 3.63·29-s − 0.463·31-s − 4.63·32-s + 4.74·34-s − 1.30·35-s − 6.00·37-s − 0.611·38-s − 1.90·40-s + 6.58·41-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 0.437·4-s + 0.447·5-s − 0.493·7-s − 0.674·8-s + 0.536·10-s + 1.67·11-s − 1.17·13-s − 0.591·14-s − 1.24·16-s + 0.678·17-s − 0.0827·19-s + 0.195·20-s + 2.00·22-s + 1.75·23-s + 0.200·25-s − 1.41·26-s − 0.215·28-s + 0.674·29-s − 0.0833·31-s − 0.820·32-s + 0.813·34-s − 0.220·35-s − 0.987·37-s − 0.0991·38-s − 0.301·40-s + 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.631406967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.631406967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 + 0.360T + 19T^{2} \) |
| 23 | \( 1 - 8.41T + 23T^{2} \) |
| 29 | \( 1 - 3.63T + 29T^{2} \) |
| 31 | \( 1 + 0.463T + 31T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 - 6.58T + 41T^{2} \) |
| 43 | \( 1 - 5.16T + 43T^{2} \) |
| 47 | \( 1 - 0.537T + 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 59 | \( 1 + 2.04T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 0.928T + 67T^{2} \) |
| 71 | \( 1 - 0.368T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 5.80T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−8.650365752733776154602597864708, −7.34868291122192259918655381957, −6.74334456014146792611307704770, −6.17836943698275328068956804225, −5.31692887423601268515184698239, −4.74318809407395925227185401130, −3.86512563360385572220300345903, −3.17849126885495465056666822559, −2.32288333952970036234236620320, −0.936425567981436582969286736818,
0.936425567981436582969286736818, 2.32288333952970036234236620320, 3.17849126885495465056666822559, 3.86512563360385572220300345903, 4.74318809407395925227185401130, 5.31692887423601268515184698239, 6.17836943698275328068956804225, 6.74334456014146792611307704770, 7.34868291122192259918655381957, 8.650365752733776154602597864708
