L(s) = 1 | − 2.76·3-s − 1.62·5-s + 4.62·9-s − 4.49·11-s + 0.103·13-s + 4.49·15-s + 2·17-s + 7.25·19-s + 5.52·23-s − 2.35·25-s − 4.49·27-s + 29-s + 6.76·31-s + 12.4·33-s + 5.25·37-s − 0.284·39-s + 5.79·41-s + 10.0·43-s − 7.52·45-s − 11.5·47-s − 7·49-s − 5.52·51-s − 7.14·53-s + 7.30·55-s − 20.0·57-s + 1.52·59-s + 9.04·61-s + ⋯ |
L(s) = 1 | − 1.59·3-s − 0.727·5-s + 1.54·9-s − 1.35·11-s + 0.0285·13-s + 1.15·15-s + 0.485·17-s + 1.66·19-s + 1.15·23-s − 0.471·25-s − 0.864·27-s + 0.185·29-s + 1.21·31-s + 2.15·33-s + 0.863·37-s − 0.0455·39-s + 0.904·41-s + 1.52·43-s − 1.12·45-s − 1.68·47-s − 49-s − 0.773·51-s − 0.982·53-s + 0.984·55-s − 2.65·57-s + 0.198·59-s + 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6297432978\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6297432978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + 2.76T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 - 0.103T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 - 9.04T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 - 1.98T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−11.26668728858203816494622074942, −10.37262160873424266073024338937, −9.559626223654177263053105795853, −7.975999286882818946689778389420, −7.39757559108079085352743982205, −6.22101831010226446872848658742, −5.30753223581452485311590370493, −4.62536170076174373980632627192, −3.07961580880794373490148983503, −0.795900739957653789871602104165,
0.795900739957653789871602104165, 3.07961580880794373490148983503, 4.62536170076174373980632627192, 5.30753223581452485311590370493, 6.22101831010226446872848658742, 7.39757559108079085352743982205, 7.975999286882818946689778389420, 9.559626223654177263053105795853, 10.37262160873424266073024338937, 11.26668728858203816494622074942