| L(s) = 1 | + 2·5-s − 4·13-s − 25-s − 16·31-s − 20·37-s + 12·41-s + 8·43-s + 10·49-s + 20·53-s − 8·65-s − 24·67-s − 32·79-s − 8·83-s − 20·89-s + 40·107-s + 18·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.10·13-s − 1/5·25-s − 2.87·31-s − 3.28·37-s + 1.87·41-s + 1.21·43-s + 10/7·49-s + 2.74·53-s − 0.992·65-s − 2.93·67-s − 3.60·79-s − 0.878·83-s − 2.11·89-s + 3.86·107-s + 1.63·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.368713514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368713514\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.995010198137142101388417422523, −8.588873024135506190080600687297, −8.543723760150973839511838206880, −7.46936644326377922499548316288, −7.33999248467100798984629511321, −7.21598547621791049459364848755, −6.99421251296758875255040990810, −5.94046592482253107494214963816, −5.92966780932420742116761318589, −5.50938060664601891246400719224, −5.38635228119104646156663736721, −4.54822255274721330288199597538, −4.38688505320406691410309751899, −3.73180961185264644448680063086, −3.42846475164819879344047876799, −2.61707077401856619644583596247, −2.43638231431957133569903812254, −1.78227186178932198842651233087, −1.44235024628130810853701511475, −0.35457355363354978229036562264,
0.35457355363354978229036562264, 1.44235024628130810853701511475, 1.78227186178932198842651233087, 2.43638231431957133569903812254, 2.61707077401856619644583596247, 3.42846475164819879344047876799, 3.73180961185264644448680063086, 4.38688505320406691410309751899, 4.54822255274721330288199597538, 5.38635228119104646156663736721, 5.50938060664601891246400719224, 5.92966780932420742116761318589, 5.94046592482253107494214963816, 6.99421251296758875255040990810, 7.21598547621791049459364848755, 7.33999248467100798984629511321, 7.46936644326377922499548316288, 8.543723760150973839511838206880, 8.588873024135506190080600687297, 8.995010198137142101388417422523
