| L(s) = 1 | + 2.09·2-s − 3-s + 2.36·4-s + 0.388·5-s − 2.09·6-s + 7-s + 0.771·8-s + 9-s + 0.811·10-s + 6.41·11-s − 2.36·12-s + 3.88·13-s + 2.09·14-s − 0.388·15-s − 3.12·16-s − 4.98·17-s + 2.09·18-s − 19-s + 0.919·20-s − 21-s + 13.4·22-s + 3.44·23-s − 0.771·24-s − 4.84·25-s + 8.12·26-s − 27-s + 2.36·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.577·3-s + 1.18·4-s + 0.173·5-s − 0.853·6-s + 0.377·7-s + 0.272·8-s + 0.333·9-s + 0.256·10-s + 1.93·11-s − 0.683·12-s + 1.07·13-s + 0.558·14-s − 0.100·15-s − 0.781·16-s − 1.20·17-s + 0.492·18-s − 0.229·19-s + 0.205·20-s − 0.218·21-s + 2.86·22-s + 0.717·23-s − 0.157·24-s − 0.969·25-s + 1.59·26-s − 0.192·27-s + 0.447·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.682986877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.682986877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 5 | \( 1 - 0.388T + 5T^{2} \) |
| 11 | \( 1 - 6.41T + 11T^{2} \) |
| 13 | \( 1 - 3.88T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 23 | \( 1 - 3.44T + 23T^{2} \) |
| 29 | \( 1 - 0.169T + 29T^{2} \) |
| 31 | \( 1 + 8.62T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 - 8.42T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 - 5.82T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 1.01T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−11.47685096821963154885527199474, −10.97232803514642848210651427701, −9.443386790050601825531005598223, −8.609016414617399720980204643918, −6.86071787260097522949000571741, −6.37544928927090577062163417151, −5.41264417966402576008142593713, −4.30434887053437066504023806442, −3.63340731583608311598654472987, −1.75514685784801947870251495642,
1.75514685784801947870251495642, 3.63340731583608311598654472987, 4.30434887053437066504023806442, 5.41264417966402576008142593713, 6.37544928927090577062163417151, 6.86071787260097522949000571741, 8.609016414617399720980204643918, 9.443386790050601825531005598223, 10.97232803514642848210651427701, 11.47685096821963154885527199474
