| L(s) = 1 | + 1.41·2-s − 3-s + 0.0121·4-s − 2.27·5-s − 1.41·6-s − 2.81·8-s + 9-s − 3.23·10-s − 2.51·11-s − 0.0121·12-s + 0.793·13-s + 2.27·15-s − 4.02·16-s + 5.45·17-s + 1.41·18-s + 3.19·19-s − 0.0275·20-s − 3.56·22-s + 0.533·23-s + 2.81·24-s + 0.187·25-s + 1.12·26-s − 27-s + 7.94·29-s + 3.23·30-s + 4.75·31-s − 0.0685·32-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.577·3-s + 0.00605·4-s − 1.01·5-s − 0.579·6-s − 0.996·8-s + 0.333·9-s − 1.02·10-s − 0.757·11-s − 0.00349·12-s + 0.220·13-s + 0.588·15-s − 1.00·16-s + 1.32·17-s + 0.334·18-s + 0.731·19-s − 0.00617·20-s − 0.760·22-s + 0.111·23-s + 0.575·24-s + 0.0374·25-s + 0.220·26-s − 0.192·27-s + 1.47·29-s + 0.589·30-s + 0.853·31-s − 0.0121·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 11 | \( 1 + 2.51T + 11T^{2} \) |
| 13 | \( 1 - 0.793T + 13T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 - 0.533T + 23T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 9.61T + 53T^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 61 | \( 1 - 8.46T + 61T^{2} \) |
| 67 | \( 1 + 6.45T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 8.93T + 89T^{2} \) |
| 97 | \( 1 - 3.67T + 97T^{2} \) |
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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
−7.57554263288694369518811330917, −7.00815451048426539132213609172, −5.91012924701234455499760838127, −5.59553844571054824785595332673, −4.70086597154428475672035796030, −4.26068576182404457299921281702, −3.31699555329675008491935562898, −2.83756041712934348850937437767, −1.13839327254742910999212539798, 0,
1.13839327254742910999212539798, 2.83756041712934348850937437767, 3.31699555329675008491935562898, 4.26068576182404457299921281702, 4.70086597154428475672035796030, 5.59553844571054824785595332673, 5.91012924701234455499760838127, 7.00815451048426539132213609172, 7.57554263288694369518811330917
