| L(s) = 1 | − 3·3-s + 5-s + 7-s + 6·9-s + 4·13-s − 3·15-s + 17-s − 3·21-s + 23-s + 25-s − 9·27-s − 6·29-s + 35-s − 3·37-s − 12·39-s + 7·41-s + 43-s + 6·45-s − 4·47-s + 49-s − 3·51-s − 11·53-s − 6·59-s + 8·61-s + 6·63-s + 4·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s + 1.10·13-s − 0.774·15-s + 0.242·17-s − 0.654·21-s + 0.208·23-s + 1/5·25-s − 1.73·27-s − 1.11·29-s + 0.169·35-s − 0.493·37-s − 1.92·39-s + 1.09·41-s + 0.152·43-s + 0.894·45-s − 0.583·47-s + 1/7·49-s − 0.420·51-s − 1.51·53-s − 0.781·59-s + 1.02·61-s + 0.755·63-s + 0.496·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.17022208863885, −12.82547112896232, −12.21771150340161, −11.85363812780247, −11.29400434010396, −11.04173545526277, −10.58676465902751, −10.26014944195238, −9.544040049437087, −9.192231040694254, −8.620890704986173, −7.822410813786791, −7.584876066963657, −6.795272833802246, −6.441996161033365, −6.015035486589539, −5.474782700464138, −5.245957279042393, −4.540927710422315, −4.109017147929161, −3.472573996463989, −2.737195341512463, −1.752900839213039, −1.454154714532483, −0.7732398155188524, 0,
0.7732398155188524, 1.454154714532483, 1.752900839213039, 2.737195341512463, 3.472573996463989, 4.109017147929161, 4.540927710422315, 5.245957279042393, 5.474782700464138, 6.015035486589539, 6.441996161033365, 6.795272833802246, 7.584876066963657, 7.822410813786791, 8.620890704986173, 9.192231040694254, 9.544040049437087, 10.26014944195238, 10.58676465902751, 11.04173545526277, 11.29400434010396, 11.85363812780247, 12.21771150340161, 12.82547112896232, 13.17022208863885
