| L(s) = 1 | − 3.16·5-s − 4.57·7-s − 2.47·11-s + 1.41·13-s + 6.47·17-s − 2.47·19-s − 5.65·23-s + 5.00·25-s − 0.333·29-s − 10.2·31-s + 14.4·35-s − 2.08·37-s − 6.47·41-s − 10.4·43-s + 13.9·49-s − 5.32·53-s + 7.81·55-s − 8.94·59-s − 10.5·61-s − 4.47·65-s − 12·67-s − 3.49·71-s − 14.9·73-s + 11.3·77-s − 1.08·79-s + 2.47·83-s − 20.4·85-s + ⋯ |
| L(s) = 1 | − 1.41·5-s − 1.72·7-s − 0.745·11-s + 0.392·13-s + 1.56·17-s − 0.567·19-s − 1.17·23-s + 1.00·25-s − 0.0619·29-s − 1.83·31-s + 2.44·35-s − 0.342·37-s − 1.01·41-s − 1.59·43-s + 1.99·49-s − 0.731·53-s + 1.05·55-s − 1.16·59-s − 1.35·61-s − 0.554·65-s − 1.46·67-s − 0.414·71-s − 1.74·73-s + 1.28·77-s − 0.121·79-s + 0.271·83-s − 2.21·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08724558664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08724558664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 0.333T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.08T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 1.08T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 4.94T + 97T^{2} \) |
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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
−7.57519930488436471515901015722, −7.27171288644084831154373004474, −6.28427316197843957597398666539, −5.84129521852587754614752787328, −4.90146571164645180681075030367, −3.96690612812597804221730770381, −3.35585855790566381604924758455, −3.09280680790657636149640816684, −1.68135219288911631158372933319, −0.13679966775258260570584844736,
0.13679966775258260570584844736, 1.68135219288911631158372933319, 3.09280680790657636149640816684, 3.35585855790566381604924758455, 3.96690612812597804221730770381, 4.90146571164645180681075030367, 5.84129521852587754614752787328, 6.28427316197843957597398666539, 7.27171288644084831154373004474, 7.57519930488436471515901015722
