| L(s) = 1 | + 2·3-s − 7-s + 9-s + 3·11-s − 13-s − 19-s − 2·21-s − 4·27-s − 3·29-s − 2·31-s + 6·33-s − 2·37-s − 2·39-s + 3·41-s − 43-s − 6·49-s + 12·53-s − 2·57-s + 6·59-s − 2·61-s − 63-s + 8·67-s + 6·71-s − 7·73-s − 3·77-s − 79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.229·19-s − 0.436·21-s − 0.769·27-s − 0.557·29-s − 0.359·31-s + 1.04·33-s − 0.328·37-s − 0.320·39-s + 0.468·41-s − 0.152·43-s − 6/7·49-s + 1.64·53-s − 0.264·57-s + 0.781·59-s − 0.256·61-s − 0.125·63-s + 0.977·67-s + 0.712·71-s − 0.819·73-s − 0.341·77-s − 0.112·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.272340572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.272340572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.15996524885580, −12.60153126131897, −12.17107463469232, −11.52385543903824, −11.30574671922915, −10.55853758681033, −10.07443529346487, −9.573050976448573, −9.223131910375769, −8.786027312150856, −8.371054497355846, −7.862528317406625, −7.284869553935277, −6.916140078884921, −6.357638656040962, −5.743849436497526, −5.310469061345768, −4.468700570114627, −4.078944626825897, −3.384522934313671, −3.257656269361021, −2.288269044046844, −2.101817857811364, −1.275120808077640, −0.4629965206681915,
0.4629965206681915, 1.275120808077640, 2.101817857811364, 2.288269044046844, 3.257656269361021, 3.384522934313671, 4.078944626825897, 4.468700570114627, 5.310469061345768, 5.743849436497526, 6.357638656040962, 6.916140078884921, 7.284869553935277, 7.862528317406625, 8.371054497355846, 8.786027312150856, 9.223131910375769, 9.573050976448573, 10.07443529346487, 10.55853758681033, 11.30574671922915, 11.52385543903824, 12.17107463469232, 12.60153126131897, 13.15996524885580
