| L(s) = 1 | − 1.22·3-s + 3.37·5-s + 3.01·7-s − 1.51·9-s + 5.36·11-s − 1.17·13-s − 4.11·15-s − 0.433·17-s + 7.85·19-s − 3.68·21-s − 7.02·23-s + 6.36·25-s + 5.50·27-s + 6.74·29-s − 2.65·31-s − 6.54·33-s + 10.1·35-s − 6.36·37-s + 1.43·39-s − 10.3·41-s − 12.1·43-s − 5.09·45-s + 8.70·47-s + 2.11·49-s + 0.529·51-s − 9.67·53-s + 18.0·55-s + ⋯ |
| L(s) = 1 | − 0.704·3-s + 1.50·5-s + 1.14·7-s − 0.503·9-s + 1.61·11-s − 0.325·13-s − 1.06·15-s − 0.105·17-s + 1.80·19-s − 0.804·21-s − 1.46·23-s + 1.27·25-s + 1.05·27-s + 1.25·29-s − 0.476·31-s − 1.14·33-s + 1.72·35-s − 1.04·37-s + 0.229·39-s − 1.62·41-s − 1.85·43-s − 0.759·45-s + 1.27·47-s + 0.302·49-s + 0.0741·51-s − 1.32·53-s + 2.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.062920111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.062920111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 - 3.01T + 7T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 + 0.433T + 17T^{2} \) |
| 19 | \( 1 - 7.85T + 19T^{2} \) |
| 23 | \( 1 + 7.02T + 23T^{2} \) |
| 29 | \( 1 - 6.74T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 + 6.36T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 + 9.67T + 53T^{2} \) |
| 59 | \( 1 - 9.68T + 59T^{2} \) |
| 61 | \( 1 + 0.450T + 61T^{2} \) |
| 67 | \( 1 - 9.11T + 67T^{2} \) |
| 71 | \( 1 - 5.15T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 9.80T + 83T^{2} \) |
| 89 | \( 1 - 1.00T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−9.753458532928045914275078069984, −8.893203866817224545613010845540, −8.132947753442399066822659824616, −6.84461150605673562429318988858, −6.28366960558853659880227804024, −5.33757515407833723897591862016, −4.96319832041371046052459507509, −3.51995996761583692802918538722, −2.06340617116138616674702651801, −1.23341106195958649786124806048,
1.23341106195958649786124806048, 2.06340617116138616674702651801, 3.51995996761583692802918538722, 4.96319832041371046052459507509, 5.33757515407833723897591862016, 6.28366960558853659880227804024, 6.84461150605673562429318988858, 8.132947753442399066822659824616, 8.893203866817224545613010845540, 9.753458532928045914275078069984
