| L(s) = 1 | + 1.73·2-s − 3-s + 1.00·4-s + 0.756·5-s − 1.73·6-s − 4.14·7-s − 1.73·8-s + 9-s + 1.31·10-s + 4.78·11-s − 1.00·12-s − 13-s − 7.17·14-s − 0.756·15-s − 4.99·16-s − 2.70·17-s + 1.73·18-s − 3.73·19-s + 0.757·20-s + 4.14·21-s + 8.28·22-s + 7.59·23-s + 1.73·24-s − 4.42·25-s − 1.73·26-s − 27-s − 4.14·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.500·4-s + 0.338·5-s − 0.707·6-s − 1.56·7-s − 0.611·8-s + 0.333·9-s + 0.414·10-s + 1.44·11-s − 0.289·12-s − 0.277·13-s − 1.91·14-s − 0.195·15-s − 1.24·16-s − 0.656·17-s + 0.408·18-s − 0.856·19-s + 0.169·20-s + 0.903·21-s + 1.76·22-s + 1.58·23-s + 0.353·24-s − 0.885·25-s − 0.339·26-s − 0.192·27-s − 0.783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.215891922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215891922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 - 0.756T + 5T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 - 7.59T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 6.95T + 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 6.31T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 5.11T + 59T^{2} \) |
| 61 | \( 1 - 2.23T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 - 1.90T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 4.41T + 83T^{2} \) |
| 89 | \( 1 - 0.167T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.669694911262952058361788954527, −7.21093529681755864155127128184, −6.50889151040199708982071095140, −6.27754074053362596194914849953, −5.56802283500780936060948981671, −4.50528132777896936260252442202, −4.06129131165451644007001730861, −3.17436893704377410993377790959, −2.33501783946100884528179186073, −0.71203783729311611282342846026,
0.71203783729311611282342846026, 2.33501783946100884528179186073, 3.17436893704377410993377790959, 4.06129131165451644007001730861, 4.50528132777896936260252442202, 5.56802283500780936060948981671, 6.27754074053362596194914849953, 6.50889151040199708982071095140, 7.21093529681755864155127128184, 8.669694911262952058361788954527
