| L(s) = 1 | − 2.33·2-s + 3.46·4-s + 1.73·5-s − 3.14·7-s − 3.42·8-s − 4.06·10-s − 1.08·11-s + 3.08·13-s + 7.36·14-s + 1.07·16-s + 7.83·17-s + 0.987·19-s + 6.02·20-s + 2.53·22-s − 1.81·23-s − 1.97·25-s − 7.21·26-s − 10.9·28-s − 8.47·29-s + 2.86·31-s + 4.33·32-s − 18.3·34-s − 5.47·35-s + 9.75·37-s − 2.30·38-s − 5.95·40-s + 6.02·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.73·4-s + 0.777·5-s − 1.19·7-s − 1.21·8-s − 1.28·10-s − 0.327·11-s + 0.855·13-s + 1.96·14-s + 0.268·16-s + 1.90·17-s + 0.226·19-s + 1.34·20-s + 0.541·22-s − 0.377·23-s − 0.395·25-s − 1.41·26-s − 2.06·28-s − 1.57·29-s + 0.514·31-s + 0.766·32-s − 3.14·34-s − 0.925·35-s + 1.60·37-s − 0.374·38-s − 0.940·40-s + 0.940·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6910781662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6910781662\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 - 7.83T + 17T^{2} \) |
| 19 | \( 1 - 0.987T + 19T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 2.86T + 31T^{2} \) |
| 37 | \( 1 - 9.75T + 37T^{2} \) |
| 41 | \( 1 - 6.02T + 41T^{2} \) |
| 43 | \( 1 - 3.81T + 43T^{2} \) |
| 47 | \( 1 - 0.987T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 71 | \( 1 + 1.02T + 71T^{2} \) |
| 73 | \( 1 - 4.74T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
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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
−10.26148419107220122372228560973, −9.651472183239803629079384286050, −9.260116265901583123049116407840, −8.067381881713533175998724270760, −7.42208205169232002871265062548, −6.24663308606354444926328549587, −5.69059417225064130259538334550, −3.62567391947467031122223068666, −2.37390552967580553805746503403, −0.947039500222732690334762210078,
0.947039500222732690334762210078, 2.37390552967580553805746503403, 3.62567391947467031122223068666, 5.69059417225064130259538334550, 6.24663308606354444926328549587, 7.42208205169232002871265062548, 8.067381881713533175998724270760, 9.260116265901583123049116407840, 9.651472183239803629079384286050, 10.26148419107220122372228560973
