| L(s) = 1 | + 2.41·3-s + 3.82·5-s + 1.41·7-s + 2.82·9-s − 11-s − 4.24·13-s + 9.24·15-s − 4.58·17-s + 0.585·19-s + 3.41·21-s + 5.58·23-s + 9.65·25-s − 0.414·27-s + 8·29-s + 8.07·31-s − 2.41·33-s + 5.41·35-s − 5.82·37-s − 10.2·39-s − 7.07·41-s − 0.828·43-s + 10.8·45-s − 4.48·47-s − 5·49-s − 11.0·51-s + 2.82·53-s − 3.82·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 1.71·5-s + 0.534·7-s + 0.942·9-s − 0.301·11-s − 1.17·13-s + 2.38·15-s − 1.11·17-s + 0.134·19-s + 0.745·21-s + 1.16·23-s + 1.93·25-s − 0.0797·27-s + 1.48·29-s + 1.44·31-s − 0.420·33-s + 0.915·35-s − 0.958·37-s − 1.64·39-s − 1.10·41-s − 0.126·43-s + 1.61·45-s − 0.654·47-s − 0.714·49-s − 1.55·51-s + 0.388·53-s − 0.516·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.598409552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.598409552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 8.07T + 31T^{2} \) |
| 37 | \( 1 + 5.82T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 0.828T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 + 6.07T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 0.828T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
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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
−9.567408487389932111274888326787, −8.723196679251823431227739172470, −8.265238748429276375857720329561, −7.10496106045072280103092405493, −6.45418485139458554603520270024, −5.18027318837337747713518095922, −4.64634246556385292204305254821, −2.99146649315298898635727298283, −2.46019208686065281319077729826, −1.58315353067553343859543527901,
1.58315353067553343859543527901, 2.46019208686065281319077729826, 2.99146649315298898635727298283, 4.64634246556385292204305254821, 5.18027318837337747713518095922, 6.45418485139458554603520270024, 7.10496106045072280103092405493, 8.265238748429276375857720329561, 8.723196679251823431227739172470, 9.567408487389932111274888326787
