| L(s) = 1 | − 2·5-s + 6·9-s + 8·19-s − 25-s − 4·29-s + 16·31-s − 12·41-s − 12·45-s − 8·59-s + 12·61-s + 24·71-s + 8·79-s + 27·81-s − 20·89-s − 16·95-s + 36·101-s − 28·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s − 32·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2·9-s + 1.83·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 1.78·45-s − 1.04·59-s + 1.53·61-s + 2.84·71-s + 0.900·79-s + 3·81-s − 2.11·89-s − 1.64·95-s + 3.58·101-s − 2.68·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.275759636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.275759636\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.05233075025307437457255620931, −9.718041539332464626372441339389, −9.664454064208399134561197183584, −9.064975301251896972903238645405, −8.257272721551026313630727910791, −8.142734267157461736751636478283, −7.73245463829029371021455980558, −7.26938529790489518941486150390, −6.82286378000839713123443864296, −6.66267580602216179562967404843, −5.97249057170617375311967569198, −5.27094420487273348104079897818, −4.85222644296536460126174026308, −4.55558651605682777273395501259, −3.75797069881104069637612385180, −3.71245667115707378202967247638, −2.97490547179958909482585564847, −2.17543261314394075770418611888, −1.38123482634235965320934201771, −0.795800109119008986207509747722,
0.795800109119008986207509747722, 1.38123482634235965320934201771, 2.17543261314394075770418611888, 2.97490547179958909482585564847, 3.71245667115707378202967247638, 3.75797069881104069637612385180, 4.55558651605682777273395501259, 4.85222644296536460126174026308, 5.27094420487273348104079897818, 5.97249057170617375311967569198, 6.66267580602216179562967404843, 6.82286378000839713123443864296, 7.26938529790489518941486150390, 7.73245463829029371021455980558, 8.142734267157461736751636478283, 8.257272721551026313630727910791, 9.064975301251896972903238645405, 9.664454064208399134561197183584, 9.718041539332464626372441339389, 10.05233075025307437457255620931
