| L(s) = 1 | + 2.50·2-s + 4.28·4-s + 0.826·5-s + 1.21·7-s + 5.72·8-s + 2.07·10-s + 11-s + 0.401·13-s + 3.04·14-s + 5.78·16-s + 0.390·17-s + 1.49·19-s + 3.54·20-s + 2.50·22-s + 3.92·23-s − 4.31·25-s + 1.00·26-s + 5.20·28-s + 3.72·29-s + 10.1·31-s + 3.05·32-s + 0.978·34-s + 1.00·35-s + 3.36·37-s + 3.73·38-s + 4.73·40-s − 4.81·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.14·4-s + 0.369·5-s + 0.459·7-s + 2.02·8-s + 0.655·10-s + 0.301·11-s + 0.111·13-s + 0.814·14-s + 1.44·16-s + 0.0946·17-s + 0.341·19-s + 0.791·20-s + 0.534·22-s + 0.817·23-s − 0.863·25-s + 0.197·26-s + 0.984·28-s + 0.692·29-s + 1.82·31-s + 0.539·32-s + 0.167·34-s + 0.169·35-s + 0.552·37-s + 0.605·38-s + 0.748·40-s − 0.752·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.650391391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.650391391\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 5 | \( 1 - 0.826T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 13 | \( 1 - 0.401T + 13T^{2} \) |
| 17 | \( 1 - 0.390T + 17T^{2} \) |
| 19 | \( 1 - 1.49T + 19T^{2} \) |
| 23 | \( 1 - 3.92T + 23T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 3.36T + 37T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 + 5.55T + 43T^{2} \) |
| 47 | \( 1 + 6.69T + 47T^{2} \) |
| 53 | \( 1 - 2.13T + 53T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 67 | \( 1 - 9.49T + 67T^{2} \) |
| 71 | \( 1 - 5.35T + 71T^{2} \) |
| 73 | \( 1 - 8.27T + 73T^{2} \) |
| 79 | \( 1 + 7.91T + 79T^{2} \) |
| 83 | \( 1 + 5.56T + 83T^{2} \) |
| 89 | \( 1 - 0.229T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.950088211982004683181412663746, −6.91999427390759224044191044242, −6.52589850193653158667486591948, −5.77978747657376786417218860023, −5.05230922448338393561660368712, −4.59856385938130947788576323700, −3.72577305041657292550067673920, −2.99974352117704956642939982302, −2.19449252215752999909600843847, −1.22516973649828535153653484165,
1.22516973649828535153653484165, 2.19449252215752999909600843847, 2.99974352117704956642939982302, 3.72577305041657292550067673920, 4.59856385938130947788576323700, 5.05230922448338393561660368712, 5.77978747657376786417218860023, 6.52589850193653158667486591948, 6.91999427390759224044191044242, 7.950088211982004683181412663746
