L(s) = 1 | + 0.944·3-s + 3.16·5-s − 2.10·9-s − 4.35·11-s + 2.82·13-s + 2.98·15-s + 7.38·17-s + 0.368·19-s − 4.56·23-s + 4.99·25-s − 4.82·27-s + 1.58·29-s + 4.76·31-s − 4.11·33-s + 7.46·37-s + 2.66·39-s + 6.92·41-s + 10.2·43-s − 6.66·45-s + 2.42·47-s + 6.97·51-s − 6.63·53-s − 13.7·55-s + 0.347·57-s + 8.27·59-s + 11.1·61-s + 8.92·65-s + ⋯ |
L(s) = 1 | + 0.545·3-s + 1.41·5-s − 0.702·9-s − 1.31·11-s + 0.783·13-s + 0.771·15-s + 1.79·17-s + 0.0844·19-s − 0.952·23-s + 0.999·25-s − 0.928·27-s + 0.294·29-s + 0.855·31-s − 0.715·33-s + 1.22·37-s + 0.427·39-s + 1.08·41-s + 1.56·43-s − 0.993·45-s + 0.353·47-s + 0.976·51-s − 0.911·53-s − 1.85·55-s + 0.0460·57-s + 1.07·59-s + 1.42·61-s + 1.10·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.808520482\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.808520482\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 0.944T + 3T^{2} \) |
| 5 | \( 1 - 3.16T + 5T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 7.38T + 17T^{2} \) |
| 19 | \( 1 - 0.368T + 19T^{2} \) |
| 23 | \( 1 + 4.56T + 23T^{2} \) |
| 29 | \( 1 - 1.58T + 29T^{2} \) |
| 31 | \( 1 - 4.76T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 2.42T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 - 8.27T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 5.02T + 67T^{2} \) |
| 71 | \( 1 - 5.15T + 71T^{2} \) |
| 73 | \( 1 + 9.32T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 6.19T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.804707624011751464051869947444, −8.059441816235958148854389156117, −7.59113020027730140081238632038, −6.20652852806891865614604861386, −5.78721650173538850498251637281, −5.21929456282538675650865117984, −3.91020123646620554381914878873, −2.81398329113189604809460544217, −2.36656212939095856816452430042, −1.06028944995060417912599424135,
1.06028944995060417912599424135, 2.36656212939095856816452430042, 2.81398329113189604809460544217, 3.91020123646620554381914878873, 5.21929456282538675650865117984, 5.78721650173538850498251637281, 6.20652852806891865614604861386, 7.59113020027730140081238632038, 8.059441816235958148854389156117, 8.804707624011751464051869947444