L(s) = 1 | + 3·3-s + 6·9-s − 5·11-s + 3·13-s − 17-s − 6·19-s + 6·23-s + 9·27-s + 9·29-s − 4·31-s − 15·33-s + 2·37-s + 9·39-s + 4·41-s − 10·43-s + 47-s − 3·51-s + 4·53-s − 18·57-s + 8·59-s − 8·61-s − 12·67-s + 18·69-s − 8·71-s + 2·73-s − 13·79-s + 9·81-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 2·9-s − 1.50·11-s + 0.832·13-s − 0.242·17-s − 1.37·19-s + 1.25·23-s + 1.73·27-s + 1.67·29-s − 0.718·31-s − 2.61·33-s + 0.328·37-s + 1.44·39-s + 0.624·41-s − 1.52·43-s + 0.145·47-s − 0.420·51-s + 0.549·53-s − 2.38·57-s + 1.04·59-s − 1.02·61-s − 1.46·67-s + 2.16·69-s − 0.949·71-s + 0.234·73-s − 1.46·79-s + 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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.32978420650380, −13.61447684961151, −13.28125525779597, −13.09871655778596, −12.54832639408093, −11.90097485495574, −10.99238841334146, −10.71937520960176, −10.20230362107169, −9.721294707105335, −8.970126167081417, −8.643631374417443, −8.301260318924382, −7.851863102060442, −7.150850870410701, −6.810556703972446, −6.029822845604533, −5.359034126117312, −4.558612685824407, −4.275298185612727, −3.454506431470042, −2.851739035616002, −2.618196704342212, −1.839116985275708, −1.163434159799124, 0,
1.163434159799124, 1.839116985275708, 2.618196704342212, 2.851739035616002, 3.454506431470042, 4.275298185612727, 4.558612685824407, 5.359034126117312, 6.029822845604533, 6.810556703972446, 7.150850870410701, 7.851863102060442, 8.301260318924382, 8.643631374417443, 8.970126167081417, 9.721294707105335, 10.20230362107169, 10.71937520960176, 10.99238841334146, 11.90097485495574, 12.54832639408093, 13.09871655778596, 13.28125525779597, 13.61447684961151, 14.32978420650380