L(s) = 1 | − 2.36·2-s + 3-s + 3.57·4-s + 3.93·5-s − 2.36·6-s − 3.72·8-s + 9-s − 9.29·10-s + 11-s + 3.57·12-s + 3.93·13-s + 3.93·15-s + 1.63·16-s − 4.72·17-s − 2.36·18-s − 4.78·19-s + 14.0·20-s − 2.36·22-s + 2.72·23-s − 3.72·24-s + 10.5·25-s − 9.29·26-s + 27-s + 7.93·29-s − 9.29·30-s − 1.15·31-s + 3.57·32-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 0.577·3-s + 1.78·4-s + 1.76·5-s − 0.964·6-s − 1.31·8-s + 0.333·9-s − 2.94·10-s + 0.301·11-s + 1.03·12-s + 1.09·13-s + 1.01·15-s + 0.409·16-s − 1.14·17-s − 0.556·18-s − 1.09·19-s + 3.14·20-s − 0.503·22-s + 0.567·23-s − 0.759·24-s + 2.10·25-s − 1.82·26-s + 0.192·27-s + 1.47·29-s − 1.69·30-s − 0.207·31-s + 0.632·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.423652853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.423652853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + 4.72T + 17T^{2} \) |
| 19 | \( 1 + 4.78T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 + 5.50T + 37T^{2} \) |
| 41 | \( 1 + 0.430T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 + 3.15T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 5.44T + 79T^{2} \) |
| 83 | \( 1 - 2.84T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−9.254148076137820904158183610331, −8.726587465341917362330223519376, −8.339291611025795272185848833627, −6.90337640803366801615183701374, −6.60536666030595618066818356139, −5.68015039869709883815369532149, −4.31283768562594336707138210583, −2.72056556367129788393911038859, −2.01360373016954531569139020812, −1.12924938233690040378552685227,
1.12924938233690040378552685227, 2.01360373016954531569139020812, 2.72056556367129788393911038859, 4.31283768562594336707138210583, 5.68015039869709883815369532149, 6.60536666030595618066818356139, 6.90337640803366801615183701374, 8.339291611025795272185848833627, 8.726587465341917362330223519376, 9.254148076137820904158183610331