| L(s) = 1 | + 2.64·2-s + 5.00·4-s + 7.93·8-s − 5.29·11-s + 11.0·16-s − 14.0·22-s + 5.29·23-s − 5·25-s − 10.5·29-s + 13.2·32-s + 6·37-s + 12·43-s − 26.4·44-s + 14.0·46-s − 13.2·50-s − 10.5·53-s − 28.0·58-s + 13.0·64-s + 4·67-s − 5.29·71-s + 15.8·74-s + 8·79-s + 31.7·86-s − 42.0·88-s + 26.4·92-s − 25.0·100-s − 28.0·106-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.50·4-s + 2.80·8-s − 1.59·11-s + 2.75·16-s − 2.98·22-s + 1.10·23-s − 25-s − 1.96·29-s + 2.33·32-s + 0.986·37-s + 1.82·43-s − 3.98·44-s + 2.06·46-s − 1.87·50-s − 1.45·53-s − 3.67·58-s + 1.62·64-s + 0.488·67-s − 0.627·71-s + 1.84·74-s + 0.900·79-s + 3.42·86-s − 4.47·88-s + 2.75·92-s − 2.50·100-s − 2.71·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.847825175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.847825175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.29T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.17895832286995930823270705118, −10.82216115514045736533181030394, −9.535213828304778577885788498129, −7.87819863379023890748178489750, −7.24600568074370316077773975368, −5.97076295085884490558763280650, −5.34158610926627434786098273374, −4.36693795227291559219296206399, −3.21673288771012501952651827597, −2.20191138215244987282741431171,
2.20191138215244987282741431171, 3.21673288771012501952651827597, 4.36693795227291559219296206399, 5.34158610926627434786098273374, 5.97076295085884490558763280650, 7.24600568074370316077773975368, 7.87819863379023890748178489750, 9.535213828304778577885788498129, 10.82216115514045736533181030394, 11.17895832286995930823270705118
