| L(s) = 1 | + 3·5-s − 4·7-s + 3·11-s + 13-s − 3·17-s − 6·19-s + 4·25-s + 3·31-s − 12·35-s + 2·37-s + 8·43-s + 2·47-s + 9·49-s + 53-s + 9·55-s − 10·59-s − 2·61-s + 3·65-s + 10·67-s + 8·71-s + 15·73-s − 12·77-s + 12·79-s − 8·83-s − 9·85-s − 5·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.51·7-s + 0.904·11-s + 0.277·13-s − 0.727·17-s − 1.37·19-s + 4/5·25-s + 0.538·31-s − 2.02·35-s + 0.328·37-s + 1.21·43-s + 0.291·47-s + 9/7·49-s + 0.137·53-s + 1.21·55-s − 1.30·59-s − 0.256·61-s + 0.372·65-s + 1.22·67-s + 0.949·71-s + 1.75·73-s − 1.36·77-s + 1.35·79-s − 0.878·83-s − 0.976·85-s − 0.529·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.69195591679536, −13.55752628148680, −12.80034860281633, −12.56349734138039, −12.19039574119005, −11.27366541114979, −10.83946469697326, −10.46662192241671, −9.667423233822527, −9.583012353856310, −9.146108217677260, −8.587052070043387, −8.048017192874739, −7.109097291654388, −6.664674102687890, −6.268325769670455, −6.088942516444666, −5.390713235982414, −4.664364129282232, −3.963269345655713, −3.643477758020671, −2.615196795472138, −2.460675210322681, −1.650651987843890, −0.9003326433265455, 0,
0.9003326433265455, 1.650651987843890, 2.460675210322681, 2.615196795472138, 3.643477758020671, 3.963269345655713, 4.664364129282232, 5.390713235982414, 6.088942516444666, 6.268325769670455, 6.664674102687890, 7.109097291654388, 8.048017192874739, 8.587052070043387, 9.146108217677260, 9.583012353856310, 9.667423233822527, 10.46662192241671, 10.83946469697326, 11.27366541114979, 12.19039574119005, 12.56349734138039, 12.80034860281633, 13.55752628148680, 13.69195591679536
