| L(s) = 1 | − 1.26·3-s + 2.94·5-s − 0.461·7-s − 1.40·9-s − 2.26·11-s − 3.36·13-s − 3.71·15-s − 5.02·17-s + 3.61·19-s + 0.582·21-s + 1.18·23-s + 3.66·25-s + 5.56·27-s − 2.59·29-s + 5.52·31-s + 2.85·33-s − 1.35·35-s − 9.01·37-s + 4.24·39-s + 4.99·41-s + 1.43·43-s − 4.14·45-s − 4.18·47-s − 6.78·49-s + 6.34·51-s + 4.69·53-s − 6.65·55-s + ⋯ |
| L(s) = 1 | − 0.728·3-s + 1.31·5-s − 0.174·7-s − 0.469·9-s − 0.681·11-s − 0.932·13-s − 0.958·15-s − 1.21·17-s + 0.829·19-s + 0.127·21-s + 0.246·23-s + 0.732·25-s + 1.07·27-s − 0.481·29-s + 0.991·31-s + 0.496·33-s − 0.229·35-s − 1.48·37-s + 0.679·39-s + 0.780·41-s + 0.218·43-s − 0.618·45-s − 0.610·47-s − 0.969·49-s + 0.887·51-s + 0.645·53-s − 0.896·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.257119447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.257119447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 + 0.461T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 + 9.01T + 37T^{2} \) |
| 41 | \( 1 - 4.99T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 + 4.18T + 47T^{2} \) |
| 53 | \( 1 - 4.69T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 0.642T + 61T^{2} \) |
| 67 | \( 1 - 4.71T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 0.176T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 4.27T + 97T^{2} \) |
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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
−7.75910969584038444978727801757, −6.92287363820343759387036991150, −6.35147634310495429783373114508, −5.75558234827685693647068914452, −5.06700581943422032582813254440, −4.73494566406213081459887318944, −3.32708719109835644871422311918, −2.53256759799696959848841176241, −1.88780352259336247367521707471, −0.55051655229408008275418399591,
0.55051655229408008275418399591, 1.88780352259336247367521707471, 2.53256759799696959848841176241, 3.32708719109835644871422311918, 4.73494566406213081459887318944, 5.06700581943422032582813254440, 5.75558234827685693647068914452, 6.35147634310495429783373114508, 6.92287363820343759387036991150, 7.75910969584038444978727801757
