| L(s) = 1 | − 2·5-s − 2·9-s + 4·13-s − 12·17-s + 3·25-s + 12·29-s + 4·37-s + 12·41-s + 4·45-s − 10·49-s − 12·53-s + 4·61-s − 8·65-s + 4·73-s − 5·81-s + 24·85-s − 12·89-s + 4·97-s + 12·101-s + 4·109-s − 12·113-s − 8·117-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2/3·9-s + 1.10·13-s − 2.91·17-s + 3/5·25-s + 2.22·29-s + 0.657·37-s + 1.87·41-s + 0.596·45-s − 1.42·49-s − 1.64·53-s + 0.512·61-s − 0.992·65-s + 0.468·73-s − 5/9·81-s + 2.60·85-s − 1.27·89-s + 0.406·97-s + 1.19·101-s + 0.383·109-s − 1.12·113-s − 0.739·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5352579714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5352579714\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−13.57191933746624476808470409800, −13.00236641577412261230757825497, −12.41085901163162733003481941315, −11.50854453196989171707939391081, −11.10973086408237495845181170606, −10.83473950301065015207539180976, −9.683565298540966130897522415203, −8.740293324354028093320034492248, −8.552217550781204179646223792184, −7.71273110823499542846308181544, −6.57891116465648258947670054106, −6.27087624192875571051851265258, −4.78130792717525308450176413839, −4.12250433368686324236236368171, −2.76929890617261215013507568311,
2.76929890617261215013507568311, 4.12250433368686324236236368171, 4.78130792717525308450176413839, 6.27087624192875571051851265258, 6.57891116465648258947670054106, 7.71273110823499542846308181544, 8.552217550781204179646223792184, 8.740293324354028093320034492248, 9.683565298540966130897522415203, 10.83473950301065015207539180976, 11.10973086408237495845181170606, 11.50854453196989171707939391081, 12.41085901163162733003481941315, 13.00236641577412261230757825497, 13.57191933746624476808470409800
