| L(s) = 1 | − 1.83·3-s + 3.97·7-s + 0.384·9-s − 2.10·11-s + 5.35·13-s + 1.29·17-s − 2.10·19-s − 7.30·21-s + 23-s + 4.81·27-s + 6.03·29-s + 8.32·31-s + 3.86·33-s + 5.10·37-s − 9.85·39-s − 8.33·41-s − 7.78·43-s − 11.3·47-s + 8.78·49-s − 2.38·51-s − 0.573·53-s + 3.86·57-s + 9.17·59-s + 13.5·61-s + 1.52·63-s + 15.8·67-s − 1.83·69-s + ⋯ |
| L(s) = 1 | − 1.06·3-s + 1.50·7-s + 0.128·9-s − 0.634·11-s + 1.48·13-s + 0.314·17-s − 0.482·19-s − 1.59·21-s + 0.208·23-s + 0.926·27-s + 1.11·29-s + 1.49·31-s + 0.673·33-s + 0.838·37-s − 1.57·39-s − 1.30·41-s − 1.18·43-s − 1.66·47-s + 1.25·49-s − 0.333·51-s − 0.0787·53-s + 0.512·57-s + 1.19·59-s + 1.73·61-s + 0.192·63-s + 1.93·67-s − 0.221·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.790562019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.790562019\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.83T + 3T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 + 2.10T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 + 2.10T + 19T^{2} \) |
| 29 | \( 1 - 6.03T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 + 8.33T + 41T^{2} \) |
| 43 | \( 1 + 7.78T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 0.573T + 53T^{2} \) |
| 59 | \( 1 - 9.17T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 8.36T + 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 + 1.51T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 0.337T + 97T^{2} \) |
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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
−8.075095727597816126653359554086, −6.66284433171686248837756576941, −6.48717744011225849679098211345, −5.48886345364431676669005723233, −5.07311051628460165561352467553, −4.50691251344857657411420935925, −3.54967056435558305630185470613, −2.53055015767278826091965064120, −1.47690135315349316136359922109, −0.74598338761876807634978197320,
0.74598338761876807634978197320, 1.47690135315349316136359922109, 2.53055015767278826091965064120, 3.54967056435558305630185470613, 4.50691251344857657411420935925, 5.07311051628460165561352467553, 5.48886345364431676669005723233, 6.48717744011225849679098211345, 6.66284433171686248837756576941, 8.075095727597816126653359554086
