| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s + 6·13-s − 2·15-s − 17-s − 2·19-s + 21-s − 25-s − 27-s + 8·29-s − 2·35-s + 2·37-s − 6·39-s + 2·41-s + 8·43-s + 2·45-s − 8·47-s + 49-s + 51-s − 2·53-s + 2·57-s + 12·59-s − 4·61-s − 63-s + 12·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.242·17-s − 0.458·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.338·35-s + 0.328·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.140·51-s − 0.274·53-s + 0.264·57-s + 1.56·59-s − 0.512·61-s − 0.125·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.671110722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671110722\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.627694822849684257986488931214, −8.810858271674775165743259094528, −8.034818988611744078902949562023, −6.74786322217944422871622167174, −6.23937732241094617204326244710, −5.61803835729553467448193127024, −4.52714021066437646502043575114, −3.52683087005780987139144003068, −2.23959892983332666617681545788, −1.00570474672251756736233046600,
1.00570474672251756736233046600, 2.23959892983332666617681545788, 3.52683087005780987139144003068, 4.52714021066437646502043575114, 5.61803835729553467448193127024, 6.23937732241094617204326244710, 6.74786322217944422871622167174, 8.034818988611744078902949562023, 8.810858271674775165743259094528, 9.627694822849684257986488931214
