| L(s) = 1 | + 2.68·3-s − 5-s − 0.508·7-s + 4.18·9-s + 5.87·11-s − 1.18·13-s − 2.68·15-s + 17-s + 0.810·19-s − 1.36·21-s − 0.508·23-s + 25-s + 3.18·27-s − 0.173·29-s + 9.06·31-s + 15.7·33-s + 0.508·35-s − 4.37·37-s − 3.18·39-s + 2·41-s − 1.36·43-s − 4.18·45-s − 10.1·47-s − 6.74·49-s + 2.68·51-s + 8.17·53-s − 5.87·55-s + ⋯ |
| L(s) = 1 | + 1.54·3-s − 0.447·5-s − 0.192·7-s + 1.39·9-s + 1.77·11-s − 0.329·13-s − 0.692·15-s + 0.242·17-s + 0.185·19-s − 0.297·21-s − 0.105·23-s + 0.200·25-s + 0.613·27-s − 0.0321·29-s + 1.62·31-s + 2.74·33-s + 0.0859·35-s − 0.719·37-s − 0.510·39-s + 0.312·41-s − 0.207·43-s − 0.624·45-s − 1.48·47-s − 0.963·49-s + 0.375·51-s + 1.12·53-s − 0.791·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.459816176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.459816176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 7 | \( 1 + 0.508T + 7T^{2} \) |
| 11 | \( 1 - 5.87T + 11T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 19 | \( 1 - 0.810T + 19T^{2} \) |
| 23 | \( 1 + 0.508T + 23T^{2} \) |
| 29 | \( 1 + 0.173T + 29T^{2} \) |
| 31 | \( 1 - 9.06T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 8.17T + 53T^{2} \) |
| 59 | \( 1 + 9.91T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 1.53T + 89T^{2} \) |
| 97 | \( 1 - 8.93T + 97T^{2} \) |
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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
−10.14375616248289959822511876344, −9.452477313972348140648822010053, −8.761193346271543882242733591958, −8.046875178135292385952588798351, −7.14713347091120147453818219014, −6.29318996037078101766858250352, −4.61160947991573309296632912841, −3.72485734907373512265634401874, −2.92922126761614873901167175469, −1.52125823993003701007384407282,
1.52125823993003701007384407282, 2.92922126761614873901167175469, 3.72485734907373512265634401874, 4.61160947991573309296632912841, 6.29318996037078101766858250352, 7.14713347091120147453818219014, 8.046875178135292385952588798351, 8.761193346271543882242733591958, 9.452477313972348140648822010053, 10.14375616248289959822511876344
