| L(s) = 1 | − 2.19·2-s + 2.83·4-s − 3.77·5-s + 0.285·7-s − 1.84·8-s + 8.29·10-s − 11-s − 4.74·13-s − 0.627·14-s − 1.61·16-s − 5.55·17-s − 19-s − 10.7·20-s + 2.19·22-s + 3.32·23-s + 9.22·25-s + 10.4·26-s + 0.809·28-s − 1.36·29-s − 7.29·31-s + 7.25·32-s + 12.2·34-s − 1.07·35-s + 2.35·37-s + 2.19·38-s + 6.96·40-s − 3.84·41-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 1.41·4-s − 1.68·5-s + 0.107·7-s − 0.652·8-s + 2.62·10-s − 0.301·11-s − 1.31·13-s − 0.167·14-s − 0.404·16-s − 1.34·17-s − 0.229·19-s − 2.39·20-s + 0.468·22-s + 0.693·23-s + 1.84·25-s + 2.04·26-s + 0.153·28-s − 0.252·29-s − 1.30·31-s + 1.28·32-s + 2.09·34-s − 0.181·35-s + 0.387·37-s + 0.356·38-s + 1.10·40-s − 0.599·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1813759194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1813759194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 - 0.285T + 7T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 5.55T + 17T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 + 3.84T + 41T^{2} \) |
| 43 | \( 1 + 0.273T + 43T^{2} \) |
| 47 | \( 1 + 5.16T + 47T^{2} \) |
| 53 | \( 1 + 2.62T + 53T^{2} \) |
| 59 | \( 1 + 0.181T + 59T^{2} \) |
| 61 | \( 1 - 4.09T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 9.56T + 71T^{2} \) |
| 73 | \( 1 + 8.53T + 73T^{2} \) |
| 79 | \( 1 - 7.80T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 - 6.83T + 89T^{2} \) |
| 97 | \( 1 - 8.58T + 97T^{2} \) |
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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
−9.068491193208793793300437057773, −8.459068716199489226362442696387, −7.73794548317040917770605386568, −7.26559388925549519948345876808, −6.62241213957627422715399468109, −4.98774914389206569846949029545, −4.31169107642192501395447778237, −3.09011057640452586626724050821, −1.95644147026161288926685763715, −0.34573254238306006299413197237,
0.34573254238306006299413197237, 1.95644147026161288926685763715, 3.09011057640452586626724050821, 4.31169107642192501395447778237, 4.98774914389206569846949029545, 6.62241213957627422715399468109, 7.26559388925549519948345876808, 7.73794548317040917770605386568, 8.459068716199489226362442696387, 9.068491193208793793300437057773
