| L(s) = 1 | + 3·5-s − 7-s + 2·11-s − 4·13-s − 5·17-s − 4·23-s + 4·25-s − 4·29-s + 8·31-s − 3·35-s + 37-s − 7·41-s − 5·43-s + 47-s + 49-s − 2·53-s + 6·55-s − 11·59-s − 14·61-s − 12·65-s + 4·67-s + 12·71-s − 2·77-s − 13·79-s + 11·83-s − 15·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 0.603·11-s − 1.10·13-s − 1.21·17-s − 0.834·23-s + 4/5·25-s − 0.742·29-s + 1.43·31-s − 0.507·35-s + 0.164·37-s − 1.09·41-s − 0.762·43-s + 0.145·47-s + 1/7·49-s − 0.274·53-s + 0.809·55-s − 1.43·59-s − 1.79·61-s − 1.48·65-s + 0.488·67-s + 1.42·71-s − 0.227·77-s − 1.46·79-s + 1.20·83-s − 1.62·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.70564694306563556386819261296, −6.68974894146957113549225918247, −6.45938910278566736703088651111, −5.64722871609286473334127656227, −4.87149072676328289509530164178, −4.16376693000873424464922263180, −3.02865666823567288408653054576, −2.24949764955529507624363655191, −1.55381783573369862120339146697, 0,
1.55381783573369862120339146697, 2.24949764955529507624363655191, 3.02865666823567288408653054576, 4.16376693000873424464922263180, 4.87149072676328289509530164178, 5.64722871609286473334127656227, 6.45938910278566736703088651111, 6.68974894146957113549225918247, 7.70564694306563556386819261296
