| L(s) = 1 | − 2·3-s + 7-s + 9-s − 3·11-s + 4·13-s − 2·19-s − 2·21-s − 3·23-s + 4·27-s + 9·29-s − 8·31-s + 6·33-s − 5·37-s − 8·39-s − 6·41-s + 11·43-s + 6·47-s + 49-s − 6·53-s + 4·57-s − 10·61-s + 63-s + 5·67-s + 6·69-s − 15·71-s + 10·73-s − 3·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 0.458·19-s − 0.436·21-s − 0.625·23-s + 0.769·27-s + 1.67·29-s − 1.43·31-s + 1.04·33-s − 0.821·37-s − 1.28·39-s − 0.937·41-s + 1.67·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.529·57-s − 1.28·61-s + 0.125·63-s + 0.610·67-s + 0.722·69-s − 1.78·71-s + 1.17·73-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.394063765603297581318741249840, −7.67882275826508571025830630483, −6.69969540415864884949227259150, −6.03920214353928435424554078591, −5.40307910060790112014258089561, −4.69313246223691679353053154699, −3.73558469990672806412796676703, −2.55591790334618826783605338806, −1.29569103530388118351699925344, 0,
1.29569103530388118351699925344, 2.55591790334618826783605338806, 3.73558469990672806412796676703, 4.69313246223691679353053154699, 5.40307910060790112014258089561, 6.03920214353928435424554078591, 6.69969540415864884949227259150, 7.67882275826508571025830630483, 8.394063765603297581318741249840
