| L(s) = 1 | + 3·3-s + 2·5-s − 4·7-s + 3·9-s + 6·11-s + 2·13-s + 6·15-s + 6·17-s − 7·19-s − 12·21-s + 4·23-s + 5·25-s − 6·29-s − 16·31-s + 18·33-s − 8·35-s + 16·37-s + 6·39-s + 3·41-s − 8·43-s + 6·45-s − 10·47-s − 2·49-s + 18·51-s + 2·53-s + 12·55-s − 21·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 1.51·7-s + 9-s + 1.80·11-s + 0.554·13-s + 1.54·15-s + 1.45·17-s − 1.60·19-s − 2.61·21-s + 0.834·23-s + 25-s − 1.11·29-s − 2.87·31-s + 3.13·33-s − 1.35·35-s + 2.63·37-s + 0.960·39-s + 0.468·41-s − 1.21·43-s + 0.894·45-s − 1.45·47-s − 2/7·49-s + 2.52·51-s + 0.274·53-s + 1.61·55-s − 2.78·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.640071234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.640071234\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 3 T - 88 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−10.00431386422429153414374754127, −9.380223639214603638906155702403, −9.252287473325924229936402473775, −8.704909694898883205662825513555, −8.670621924200620095138294850563, −7.85957994609888695610652195154, −7.67501456563732144815196020146, −6.98010381012137288010553702983, −6.54428525324287580171982652725, −6.32503752002096581382086677644, −5.93937603347040367790963498606, −5.36545148723838319832003762488, −4.73229448925364767081950572092, −3.93056353121384108972772128180, −3.54496741903122418953904651114, −3.41075162547430933125434319580, −2.89369548827754809882945450080, −2.06183208017385057947220202493, −1.83239589263981842342506649680, −0.840316365765853048516606605503,
0.840316365765853048516606605503, 1.83239589263981842342506649680, 2.06183208017385057947220202493, 2.89369548827754809882945450080, 3.41075162547430933125434319580, 3.54496741903122418953904651114, 3.93056353121384108972772128180, 4.73229448925364767081950572092, 5.36545148723838319832003762488, 5.93937603347040367790963498606, 6.32503752002096581382086677644, 6.54428525324287580171982652725, 6.98010381012137288010553702983, 7.67501456563732144815196020146, 7.85957994609888695610652195154, 8.670621924200620095138294850563, 8.704909694898883205662825513555, 9.252287473325924229936402473775, 9.380223639214603638906155702403, 10.00431386422429153414374754127
