| L(s) = 1 | − 2·7-s + 6·11-s + 13-s + 19-s + 6·23-s + 3·29-s − 8·31-s − 37-s + 9·41-s − 8·43-s − 3·47-s − 3·49-s − 3·53-s − 6·59-s − 10·61-s + 13·67-s + 9·71-s − 4·73-s − 12·77-s − 11·79-s + 12·83-s + 6·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.80·11-s + 0.277·13-s + 0.229·19-s + 1.25·23-s + 0.557·29-s − 1.43·31-s − 0.164·37-s + 1.40·41-s − 1.21·43-s − 0.437·47-s − 3/7·49-s − 0.412·53-s − 0.781·59-s − 1.28·61-s + 1.58·67-s + 1.06·71-s − 0.468·73-s − 1.36·77-s − 1.23·79-s + 1.31·83-s + 0.635·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.491417035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.491417035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.59502490147825, −14.11560400193585, −13.63925474783846, −12.91033318721297, −12.64437396906349, −12.06114777710786, −11.42225211171160, −11.10313250203081, −10.48846206929434, −9.668600638458168, −9.405734034512672, −8.953642490384527, −8.408091525955990, −7.602171245339063, −7.028093816767445, −6.528748981986207, −6.179101917443573, −5.431455481521163, −4.717119098245104, −4.115190560565334, −3.344485304426403, −3.180004086620502, −2.023903950083853, −1.355613004464540, −0.5977354772602166,
0.5977354772602166, 1.355613004464540, 2.023903950083853, 3.180004086620502, 3.344485304426403, 4.115190560565334, 4.717119098245104, 5.431455481521163, 6.179101917443573, 6.528748981986207, 7.028093816767445, 7.602171245339063, 8.408091525955990, 8.953642490384527, 9.405734034512672, 9.668600638458168, 10.48846206929434, 11.10313250203081, 11.42225211171160, 12.06114777710786, 12.64437396906349, 12.91033318721297, 13.63925474783846, 14.11560400193585, 14.59502490147825
