| L(s) = 1 | − 2-s − 4-s + 2.93·5-s + 1.53·7-s + 3·8-s − 2.93·10-s + 2.16·11-s + 4.24·13-s − 1.53·14-s − 16-s − 2.82·19-s − 2.93·20-s − 2.16·22-s + 8.92·23-s + 3.58·25-s − 4.24·26-s − 1.53·28-s + 0.317·29-s + 3.69·31-s − 5·32-s + 4.48·35-s + 9.68·37-s + 2.82·38-s + 8.79·40-s + 0.317·41-s + 5.65·43-s − 2.16·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.5·4-s + 1.31·5-s + 0.578·7-s + 1.06·8-s − 0.926·10-s + 0.652·11-s + 1.17·13-s − 0.409·14-s − 0.250·16-s − 0.648·19-s − 0.655·20-s − 0.461·22-s + 1.86·23-s + 0.717·25-s − 0.832·26-s − 0.289·28-s + 0.0588·29-s + 0.663·31-s − 0.883·32-s + 0.758·35-s + 1.59·37-s + 0.458·38-s + 1.38·40-s + 0.0495·41-s + 0.862·43-s − 0.326·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.770942619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.770942619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 - 0.317T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 - 9.68T + 37T^{2} \) |
| 41 | \( 1 - 0.317T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.92T + 71T^{2} \) |
| 73 | \( 1 - 3.82T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 2.48T + 97T^{2} \) |
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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
−8.993799659457499758346106499696, −8.371155337476458636084124031929, −7.53926330564919561050125070932, −6.47851006679254370843630862836, −5.93412208411902139274732195881, −4.90648667999292764884640040598, −4.28346123551140393892427555888, −2.99649124167832676873437049150, −1.68739395980479503988778820148, −1.07350905354962134852073729317,
1.07350905354962134852073729317, 1.68739395980479503988778820148, 2.99649124167832676873437049150, 4.28346123551140393892427555888, 4.90648667999292764884640040598, 5.93412208411902139274732195881, 6.47851006679254370843630862836, 7.53926330564919561050125070932, 8.371155337476458636084124031929, 8.993799659457499758346106499696
