L(s) = 1 | + 1.81·3-s + 2.70·5-s − 3.36·7-s + 0.298·9-s + 5.17·11-s + 13-s + 4.90·15-s + 6.70·17-s − 5.17·19-s − 6.10·21-s + 2.29·25-s − 4.90·27-s − 2·29-s − 8.80·31-s + 9.40·33-s − 9.08·35-s + 2.70·37-s + 1.81·39-s + 3.40·41-s + 8.53·43-s + 0.806·45-s − 3.36·47-s + 4.29·49-s + 12.1·51-s − 11.4·53-s + 13.9·55-s − 9.40·57-s + ⋯ |
L(s) = 1 | + 1.04·3-s + 1.20·5-s − 1.27·7-s + 0.0994·9-s + 1.56·11-s + 0.277·13-s + 1.26·15-s + 1.62·17-s − 1.18·19-s − 1.33·21-s + 0.459·25-s − 0.944·27-s − 0.371·29-s − 1.58·31-s + 1.63·33-s − 1.53·35-s + 0.444·37-s + 0.290·39-s + 0.531·41-s + 1.30·43-s + 0.120·45-s − 0.490·47-s + 0.614·49-s + 1.70·51-s − 1.56·53-s + 1.88·55-s − 1.24·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.117534726\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.117534726\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 1.81T + 3T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 + 5.17T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8.80T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 - 8.53T + 43T^{2} \) |
| 47 | \( 1 + 3.36T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 3.09T + 79T^{2} \) |
| 83 | \( 1 + 1.54T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−11.07346018747580726469529385500, −9.869216570009413572272856148872, −9.401271125912948868648489118613, −8.825347872487271081578967918482, −7.53855917356795547010575450045, −6.32477399448814862735134204577, −5.81769980246510077698773678534, −3.90867891403886856696370633029, −3.05473047718636732014897962769, −1.73478642937009357582373617156,
1.73478642937009357582373617156, 3.05473047718636732014897962769, 3.90867891403886856696370633029, 5.81769980246510077698773678534, 6.32477399448814862735134204577, 7.53855917356795547010575450045, 8.825347872487271081578967918482, 9.401271125912948868648489118613, 9.869216570009413572272856148872, 11.07346018747580726469529385500