| L(s) = 1 | + 3·3-s − 3·5-s + 2·7-s + 6·9-s + 11-s − 9·15-s − 6·17-s − 4·19-s + 6·21-s − 23-s + 4·25-s + 9·27-s − 8·29-s + 7·31-s + 3·33-s − 6·35-s − 37-s + 4·41-s − 6·43-s − 18·45-s + 8·47-s − 3·49-s − 18·51-s + 2·53-s − 3·55-s − 12·57-s + 59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s + 0.301·11-s − 2.32·15-s − 1.45·17-s − 0.917·19-s + 1.30·21-s − 0.208·23-s + 4/5·25-s + 1.73·27-s − 1.48·29-s + 1.25·31-s + 0.522·33-s − 1.01·35-s − 0.164·37-s + 0.624·41-s − 0.914·43-s − 2.68·45-s + 1.16·47-s − 3/7·49-s − 2.52·51-s + 0.274·53-s − 0.404·55-s − 1.58·57-s + 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655423653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655423653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−12.89485266934258261284103101270, −11.72418487570803444661114692543, −10.77878158389799382775665542910, −9.318682460625456916246304335752, −8.441695809340897861092685799301, −7.910918321149116453263180886740, −6.87292908671921502002689767413, −4.48090140750080174715782907716, −3.74300055016635814217609152843, −2.20819379597331931749827928065,
2.20819379597331931749827928065, 3.74300055016635814217609152843, 4.48090140750080174715782907716, 6.87292908671921502002689767413, 7.910918321149116453263180886740, 8.441695809340897861092685799301, 9.318682460625456916246304335752, 10.77878158389799382775665542910, 11.72418487570803444661114692543, 12.89485266934258261284103101270
