| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s + 11-s + 6·17-s − 4·19-s + 6·21-s + 23-s − 9·27-s − 8·29-s + 7·31-s − 3·33-s + 37-s + 4·41-s + 6·43-s − 8·47-s − 3·49-s − 18·51-s − 2·53-s + 12·57-s + 59-s + 4·61-s − 12·63-s − 5·67-s − 3·69-s − 3·71-s − 16·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s + 0.301·11-s + 1.45·17-s − 0.917·19-s + 1.30·21-s + 0.208·23-s − 1.73·27-s − 1.48·29-s + 1.25·31-s − 0.522·33-s + 0.164·37-s + 0.624·41-s + 0.914·43-s − 1.16·47-s − 3/7·49-s − 2.52·51-s − 0.274·53-s + 1.58·57-s + 0.130·59-s + 0.512·61-s − 1.51·63-s − 0.610·67-s − 0.361·69-s − 0.356·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7403279640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7403279640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 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
−8.233734369869504902321993259187, −7.38482277010609614505189378697, −6.72333400475352312153017747863, −6.01984015558714566029316036157, −5.66850565381481627835370942443, −4.73528892750629836914255376212, −4.01680477399382510486658042347, −3.02758483437257834258102449051, −1.58549778325340610118173492902, −0.54792796942210733028996686469,
0.54792796942210733028996686469, 1.58549778325340610118173492902, 3.02758483437257834258102449051, 4.01680477399382510486658042347, 4.73528892750629836914255376212, 5.66850565381481627835370942443, 6.01984015558714566029316036157, 6.72333400475352312153017747863, 7.38482277010609614505189378697, 8.233734369869504902321993259187
