| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s + 11-s − 6·13-s + 5·16-s + 5·17-s − 7·19-s − 6·20-s + 2·22-s + 4·23-s + 5·25-s − 12·26-s − 4·29-s + 12·31-s + 6·32-s + 10·34-s − 2·37-s − 14·38-s − 8·40-s − 3·41-s + 43-s + 3·44-s + 8·46-s + 10·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s + 0.301·11-s − 1.66·13-s + 5/4·16-s + 1.21·17-s − 1.60·19-s − 1.34·20-s + 0.426·22-s + 0.834·23-s + 25-s − 2.35·26-s − 0.742·29-s + 2.15·31-s + 1.06·32-s + 1.71·34-s − 0.328·37-s − 2.27·38-s − 1.26·40-s − 0.468·41-s + 0.152·43-s + 0.452·44-s + 1.17·46-s + 1.41·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.009958276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.009958276\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.824537020284113034576864295785, −8.605883999802532178731957389133, −8.184883916916409380258534138285, −7.77931901657701262305213377583, −7.46953214515554653338834033136, −6.98116186406118996901187215024, −6.71160211739112702738620937727, −6.49450195190736478250962917695, −5.88423158706739244196362519730, −5.34229888889911994662927712937, −4.98605236920549854540505505085, −4.89783466270173034517463189919, −4.21424714782467516569326268032, −3.87465620460471598696718811595, −3.62518222537799671455367773040, −2.93773597615254736522531702806, −2.46873780427177624427120869078, −2.28054833530608926522103083145, −1.29618045355674907519473690409, −0.59523808368005027353619726588,
0.59523808368005027353619726588, 1.29618045355674907519473690409, 2.28054833530608926522103083145, 2.46873780427177624427120869078, 2.93773597615254736522531702806, 3.62518222537799671455367773040, 3.87465620460471598696718811595, 4.21424714782467516569326268032, 4.89783466270173034517463189919, 4.98605236920549854540505505085, 5.34229888889911994662927712937, 5.88423158706739244196362519730, 6.49450195190736478250962917695, 6.71160211739112702738620937727, 6.98116186406118996901187215024, 7.46953214515554653338834033136, 7.77931901657701262305213377583, 8.184883916916409380258534138285, 8.605883999802532178731957389133, 8.824537020284113034576864295785
