| L(s) = 1 | − 2·3-s + 9-s + 4·11-s − 2·13-s + 8·17-s − 6·19-s + 4·23-s + 4·27-s + 6·29-s − 4·31-s − 8·33-s − 10·37-s + 4·39-s − 4·41-s + 4·43-s + 4·47-s − 16·51-s + 10·53-s + 12·57-s − 14·59-s − 10·61-s − 4·67-s − 8·69-s + 12·71-s + 4·73-s + 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 1.94·17-s − 1.37·19-s + 0.834·23-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 1.39·33-s − 1.64·37-s + 0.640·39-s − 0.624·41-s + 0.609·43-s + 0.583·47-s − 2.24·51-s + 1.37·53-s + 1.58·57-s − 1.82·59-s − 1.28·61-s − 0.488·67-s − 0.963·69-s + 1.42·71-s + 0.468·73-s + 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.400476989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400476989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.94156398891818, −13.72090269672926, −12.61196374210309, −12.40456383678401, −12.09865903015554, −11.68923243921715, −10.89143333331489, −10.65363178385778, −10.18844256132419, −9.517006065617996, −9.005416936789094, −8.493659042273580, −7.863688321468506, −7.136778278852986, −6.777333503984918, −6.235803096217869, −5.700085530613945, −5.214685538167949, −4.689166386654042, −4.042672231993798, −3.390804191730533, −2.778314932810430, −1.792324186871444, −1.169375849075338, −0.4664057562097273,
0.4664057562097273, 1.169375849075338, 1.792324186871444, 2.778314932810430, 3.390804191730533, 4.042672231993798, 4.689166386654042, 5.214685538167949, 5.700085530613945, 6.235803096217869, 6.777333503984918, 7.136778278852986, 7.863688321468506, 8.493659042273580, 9.005416936789094, 9.517006065617996, 10.18844256132419, 10.65363178385778, 10.89143333331489, 11.68923243921715, 12.09865903015554, 12.40456383678401, 12.61196374210309, 13.72090269672926, 13.94156398891818
