| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 4·13-s + 2·15-s + 21-s − 6·23-s − 25-s − 27-s − 4·29-s + 2·35-s + 2·37-s − 4·39-s − 10·41-s + 4·43-s − 2·45-s + 12·47-s + 49-s + 6·53-s + 10·59-s − 10·61-s − 63-s − 8·65-s + 12·67-s + 6·69-s + 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s + 0.218·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.742·29-s + 0.338·35-s + 0.328·37-s − 0.640·39-s − 1.56·41-s + 0.609·43-s − 0.298·45-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 1.30·59-s − 1.28·61-s − 0.125·63-s − 0.992·65-s + 1.46·67-s + 0.722·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9886231310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9886231310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.71715766745043, −13.32901515747465, −12.69685120110393, −12.15125823356192, −11.95509201215149, −11.25015218699797, −10.99728621979824, −10.44231138077689, −9.843307339391581, −9.426362296716137, −8.675593139553334, −8.275456460045319, −7.775663636077433, −7.199299994722051, −6.683289505568164, −6.135409302481452, −5.627347555012002, −5.157851896475231, −4.260053508538469, −3.799708607528197, −3.648105890757221, −2.616777834902879, −1.939374797772499, −1.084152884697928, −0.3709748746206117,
0.3709748746206117, 1.084152884697928, 1.939374797772499, 2.616777834902879, 3.648105890757221, 3.799708607528197, 4.260053508538469, 5.157851896475231, 5.627347555012002, 6.135409302481452, 6.683289505568164, 7.199299994722051, 7.775663636077433, 8.275456460045319, 8.675593139553334, 9.426362296716137, 9.843307339391581, 10.44231138077689, 10.99728621979824, 11.25015218699797, 11.95509201215149, 12.15125823356192, 12.69685120110393, 13.32901515747465, 13.71715766745043
