L(s) = 1 | + 2-s − 3-s + 4-s − 3.41·5-s − 6-s + 8-s + 9-s − 3.41·10-s − 0.298·11-s − 12-s − 4.72·13-s + 3.41·15-s + 16-s − 3.54·17-s + 18-s + 7.98·19-s − 3.41·20-s − 0.298·22-s + 23-s − 24-s + 6.65·25-s − 4.72·26-s − 27-s + 8.03·29-s + 3.41·30-s + 9.98·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.52·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.07·10-s − 0.0900·11-s − 0.288·12-s − 1.30·13-s + 0.881·15-s + 0.250·16-s − 0.860·17-s + 0.235·18-s + 1.83·19-s − 0.763·20-s − 0.0636·22-s + 0.208·23-s − 0.204·24-s + 1.33·25-s − 0.925·26-s − 0.192·27-s + 1.49·29-s + 0.623·30-s + 1.79·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 0.298T + 11T^{2} \) |
| 13 | \( 1 + 4.72T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 31 | \( 1 - 9.98T + 31T^{2} \) |
| 37 | \( 1 + 3.54T + 37T^{2} \) |
| 41 | \( 1 - 0.817T + 41T^{2} \) |
| 43 | \( 1 + 6.99T + 43T^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 5.57T + 71T^{2} \) |
| 73 | \( 1 + 0.776T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 0.884T + 83T^{2} \) |
| 89 | \( 1 + 2.43T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.51989787609742264766034394864, −6.88983778568427138192221645853, −6.34214738087194059957063643217, −5.06587564701701859870859289862, −4.89940380347789854185914907135, −4.15827723038528593900367427539, −3.24416573618782249956792631155, −2.63248154481851124218805999272, −1.14305165865844566826536643843, 0,
1.14305165865844566826536643843, 2.63248154481851124218805999272, 3.24416573618782249956792631155, 4.15827723038528593900367427539, 4.89940380347789854185914907135, 5.06587564701701859870859289862, 6.34214738087194059957063643217, 6.88983778568427138192221645853, 7.51989787609742264766034394864