| L(s) = 1 | − 2-s − 2·5-s + 8-s + 2·10-s + 11-s − 6·13-s − 16-s − 10·17-s + 14·19-s − 22-s + 4·23-s + 5·25-s + 6·26-s − 4·29-s − 6·31-s + 10·34-s + 4·37-s − 14·38-s − 2·40-s − 3·41-s + 43-s − 4·46-s − 5·50-s − 24·53-s − 2·55-s + 4·58-s + 7·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.301·11-s − 1.66·13-s − 1/4·16-s − 2.42·17-s + 3.21·19-s − 0.213·22-s + 0.834·23-s + 25-s + 1.17·26-s − 0.742·29-s − 1.07·31-s + 1.71·34-s + 0.657·37-s − 2.27·38-s − 0.316·40-s − 0.468·41-s + 0.152·43-s − 0.589·46-s − 0.707·50-s − 3.29·53-s − 0.269·55-s + 0.525·58-s + 0.911·59-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
| \(L(1)\) |
\(\approx\) |
\(0.1022440464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1022440464\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 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 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.389032723250591092267361799915, −8.597001314980560310076659636193, −8.482167070954187855283316261515, −7.64544554329000698447610735332, −7.56765898186647019957323494429, −7.33930056864690403149829449754, −7.05524617098335344162528340343, −6.43312372774243194834876687147, −6.19317360826032484676551970370, −5.31946472529751217489842616550, −5.02005326524542020550541820172, −4.88047222258114057166648986105, −4.36905592984673943753994428568, −3.78541404618012187678545809292, −3.39611634723352306465771727152, −2.72915644960097711265401061688, −2.57408066638272829838514521048, −1.59107382343939269847619266726, −1.21452442512922230404611134347, −0.12773854479166654269510290279,
0.12773854479166654269510290279, 1.21452442512922230404611134347, 1.59107382343939269847619266726, 2.57408066638272829838514521048, 2.72915644960097711265401061688, 3.39611634723352306465771727152, 3.78541404618012187678545809292, 4.36905592984673943753994428568, 4.88047222258114057166648986105, 5.02005326524542020550541820172, 5.31946472529751217489842616550, 6.19317360826032484676551970370, 6.43312372774243194834876687147, 7.05524617098335344162528340343, 7.33930056864690403149829449754, 7.56765898186647019957323494429, 7.64544554329000698447610735332, 8.482167070954187855283316261515, 8.597001314980560310076659636193, 9.389032723250591092267361799915
