L(s) = 1 | + 2-s + 3-s + 4-s + 3.41·5-s + 6-s + 8-s + 9-s + 3.41·10-s + 2.29·11-s + 12-s − 1.54·13-s + 3.41·15-s + 16-s − 2.72·17-s + 18-s + 7.15·19-s + 3.41·20-s + 2.29·22-s + 23-s + 24-s + 6.65·25-s − 1.54·26-s + 27-s − 9.69·29-s + 3.41·30-s + 5.15·31-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 + 0.693·11-s + 0.288·12-s − 0.429·13-s + 0.881·15-s + 0.250·16-s − 0.659·17-s + 0.235·18-s + 1.64·19-s + 0.763·20-s + 0.490·22-s + 0.208·23-s + 0.204·24-s + 1.33·25-s − 0.303·26-s + 0.192·27-s − 1.80·29-s + 0.623·30-s + 0.925·31-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
\(L(1)\) |
\(\approx\) |
\(6.033997767\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.033997767\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 + 2.72T + 17T^{2} \) |
| 19 | \( 1 - 7.15T + 19T^{2} \) |
| 29 | \( 1 + 9.69T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 + 6.01T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 0.557T + 47T^{2} \) |
| 53 | \( 1 + 3.31T + 53T^{2} \) |
| 59 | \( 1 + 7.69T + 59T^{2} \) |
| 61 | \( 1 + 0.0113T + 61T^{2} \) |
| 67 | \( 1 - 8.93T + 67T^{2} \) |
| 71 | \( 1 + 9.25T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 1.71T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 5.76T + 97T^{2} \) |
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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
−7.74149029726467905850673671157, −7.22911386728837655485713169545, −6.41303653586104001257413354832, −5.85102745134566625366883429045, −5.17185446719591091642677395850, −4.43856349247759557483441101380, −3.49212564280571627776146902271, −2.72839653219958247413428148072, −1.99726765775455513003810426827, −1.22080431292357361817214059081,
1.22080431292357361817214059081, 1.99726765775455513003810426827, 2.72839653219958247413428148072, 3.49212564280571627776146902271, 4.43856349247759557483441101380, 5.17185446719591091642677395850, 5.85102745134566625366883429045, 6.41303653586104001257413354832, 7.22911386728837655485713169545, 7.74149029726467905850673671157