| L(s) = 1 | + 2-s + 4-s − 1.41·5-s + 8-s − 1.41·10-s − 6.24·11-s + 3·13-s + 16-s + 1.58·17-s + 7.41·19-s − 1.41·20-s − 6.24·22-s − 23-s − 2.99·25-s + 3·26-s + 3.24·29-s + 7.24·31-s + 32-s + 1.58·34-s + 6.48·37-s + 7.41·38-s − 1.41·40-s + 2.82·41-s − 5.24·43-s − 6.24·44-s − 46-s + 7.07·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.632·5-s + 0.353·8-s − 0.447·10-s − 1.88·11-s + 0.832·13-s + 0.250·16-s + 0.384·17-s + 1.70·19-s − 0.316·20-s − 1.33·22-s − 0.208·23-s − 0.599·25-s + 0.588·26-s + 0.602·29-s + 1.30·31-s + 0.176·32-s + 0.271·34-s + 1.06·37-s + 1.20·38-s − 0.223·40-s + 0.441·41-s − 0.799·43-s − 0.941·44-s − 0.147·46-s + 1.03·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.448592657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.448592657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 7.41T + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 - 2.75T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 4.75T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 5.31T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 9.17T + 97T^{2} \) |
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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
−8.622644129773862539529832504706, −7.79379376956131483533436271788, −7.59407855485482138496048055403, −6.43294557206771289998648121994, −5.58091837216949866736706019275, −5.02328257322575875980958178394, −4.05524389291606364540711530746, −3.19336032846081947983380085263, −2.46674086760496643756866727522, −0.890384696189987934251088519419,
0.890384696189987934251088519419, 2.46674086760496643756866727522, 3.19336032846081947983380085263, 4.05524389291606364540711530746, 5.02328257322575875980958178394, 5.58091837216949866736706019275, 6.43294557206771289998648121994, 7.59407855485482138496048055403, 7.79379376956131483533436271788, 8.622644129773862539529832504706
