| L(s) = 1 | − 2·5-s − 3·7-s + 2·11-s − 13-s + 6·17-s + 4·23-s − 25-s − 2·29-s + 7·31-s + 6·35-s + 37-s + 8·41-s + 7·43-s − 8·47-s + 2·49-s + 8·53-s − 4·55-s + 12·59-s − 5·61-s + 2·65-s + 9·67-s + 2·71-s − 15·73-s − 6·77-s + 11·79-s + 6·83-s − 12·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s + 0.603·11-s − 0.277·13-s + 1.45·17-s + 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.25·31-s + 1.01·35-s + 0.164·37-s + 1.24·41-s + 1.06·43-s − 1.16·47-s + 2/7·49-s + 1.09·53-s − 0.539·55-s + 1.56·59-s − 0.640·61-s + 0.248·65-s + 1.09·67-s + 0.237·71-s − 1.75·73-s − 0.683·77-s + 1.23·79-s + 0.658·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.14973357636813, −12.77250042881465, −12.19988208422958, −12.00479231740176, −11.44243895588968, −11.04401368265229, −10.34109624850865, −9.916400499808196, −9.588586843926883, −9.036429220346662, −8.578322853974921, −7.859260494497055, −7.613964007507224, −7.110586416356708, −6.458902704504911, −6.197732048280911, −5.480129236032349, −5.020322740817970, −4.262595185425090, −3.807758570345135, −3.443474443284669, −2.813870918708242, −2.345796449399397, −1.214950169587394, −0.8211778709417679, 0,
0.8211778709417679, 1.214950169587394, 2.345796449399397, 2.813870918708242, 3.443474443284669, 3.807758570345135, 4.262595185425090, 5.020322740817970, 5.480129236032349, 6.197732048280911, 6.458902704504911, 7.110586416356708, 7.613964007507224, 7.859260494497055, 8.578322853974921, 9.036429220346662, 9.588586843926883, 9.916400499808196, 10.34109624850865, 11.04401368265229, 11.44243895588968, 12.00479231740176, 12.19988208422958, 12.77250042881465, 13.14973357636813
