| L(s) = 1 | − 2.37·3-s + 0.792·5-s − 3.31·7-s + 2.62·9-s − 3.37·11-s − 4.10·13-s − 1.87·15-s + 5·17-s − 19-s + 7.86·21-s − 2.52·23-s − 4.37·25-s + 0.883·27-s − 5.98·29-s − 3.46·31-s + 8·33-s − 2.62·35-s + 3.16·37-s + 9.74·39-s − 9.37·43-s + 2.08·45-s − 9.30·47-s + 4·49-s − 11.8·51-s − 9.45·53-s − 2.67·55-s + 2.37·57-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.354·5-s − 1.25·7-s + 0.875·9-s − 1.01·11-s − 1.13·13-s − 0.485·15-s + 1.21·17-s − 0.229·19-s + 1.71·21-s − 0.526·23-s − 0.874·25-s + 0.169·27-s − 1.11·29-s − 0.622·31-s + 1.39·33-s − 0.444·35-s + 0.521·37-s + 1.56·39-s − 1.42·43-s + 0.310·45-s − 1.35·47-s + 0.571·49-s − 1.66·51-s − 1.29·53-s − 0.360·55-s + 0.314·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2273369603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2273369603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 0.792T + 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 + 5.98T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 + 9.30T + 47T^{2} \) |
| 53 | \( 1 + 9.45T + 53T^{2} \) |
| 59 | \( 1 + 4.37T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 - 8.74T + 97T^{2} \) |
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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
−7.954158076150711672234102200960, −7.54029201891683860598208806294, −6.49453796754709100540589275304, −6.17061060415983014602291312778, −5.28764080021510031502752826886, −5.01518394144020839709300150338, −3.72950225347359768855410841782, −2.91334176843932486760761107150, −1.82312604078891886191795958705, −0.26618321575937428634450680995,
0.26618321575937428634450680995, 1.82312604078891886191795958705, 2.91334176843932486760761107150, 3.72950225347359768855410841782, 5.01518394144020839709300150338, 5.28764080021510031502752826886, 6.17061060415983014602291312778, 6.49453796754709100540589275304, 7.54029201891683860598208806294, 7.954158076150711672234102200960
