L(s) = 1 | + 3-s − 5-s − 4.46·7-s + 9-s − 3.39·11-s − 4.70·13-s − 15-s − 6.54·17-s + 6.22·19-s − 4.46·21-s − 23-s + 25-s + 27-s + 0.448·29-s + 5.15·31-s − 3.39·33-s + 4.46·35-s − 5.98·37-s − 4.70·39-s + 3.07·41-s + 4.47·43-s − 45-s − 6.22·47-s + 12.9·49-s − 6.54·51-s − 2.59·53-s + 3.39·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.68·7-s + 0.333·9-s − 1.02·11-s − 1.30·13-s − 0.258·15-s − 1.58·17-s + 1.42·19-s − 0.974·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.0832·29-s + 0.925·31-s − 0.590·33-s + 0.754·35-s − 0.984·37-s − 0.753·39-s + 0.479·41-s + 0.683·43-s − 0.149·45-s − 0.908·47-s + 1.84·49-s − 0.916·51-s − 0.355·53-s + 0.457·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8654461534\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8654461534\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 4.46T + 7T^{2} \) |
| 11 | \( 1 + 3.39T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 - 6.22T + 19T^{2} \) |
| 29 | \( 1 - 0.448T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 + 5.98T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 + 4.91T + 61T^{2} \) |
| 67 | \( 1 + 6.08T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 + 8.70T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 4.30T + 97T^{2} \) |
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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
−8.005510937914596088162700017536, −7.46799117326591776416891949987, −6.84640733730721831575239808709, −6.17676926113442828921726172439, −5.12704350304752739576578327657, −4.47986372596935319762172090626, −3.44149420932361365039407454792, −2.89194634994068628803332016180, −2.21802730382417712860980062184, −0.44719517368359605850515577495,
0.44719517368359605850515577495, 2.21802730382417712860980062184, 2.89194634994068628803332016180, 3.44149420932361365039407454792, 4.47986372596935319762172090626, 5.12704350304752739576578327657, 6.17676926113442828921726172439, 6.84640733730721831575239808709, 7.46799117326591776416891949987, 8.005510937914596088162700017536