| L(s) = 1 | − 3-s + 9-s − 6·11-s + 6·13-s − 4·19-s − 27-s − 8·29-s + 2·31-s + 6·33-s − 4·37-s − 6·39-s + 10·41-s − 6·43-s + 2·47-s − 10·53-s + 4·57-s − 4·59-s − 14·61-s + 14·67-s − 8·71-s + 6·73-s + 8·79-s + 81-s + 8·83-s + 8·87-s + 18·89-s − 2·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s + 1.66·13-s − 0.917·19-s − 0.192·27-s − 1.48·29-s + 0.359·31-s + 1.04·33-s − 0.657·37-s − 0.960·39-s + 1.56·41-s − 0.914·43-s + 0.291·47-s − 1.37·53-s + 0.529·57-s − 0.520·59-s − 1.79·61-s + 1.71·67-s − 0.949·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s + 0.878·83-s + 0.857·87-s + 1.90·89-s − 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.64212896273003, −13.94727152609605, −13.41514023970356, −13.06734600125655, −12.68338608410251, −12.13165520614327, −11.28875544921153, −11.02908923178337, −10.59718188851624, −10.24241899311301, −9.397435342591218, −8.964884723188614, −8.247093816047277, −7.853156987653098, −7.377872702278671, −6.540215605882846, −6.045056732645038, −5.718474132690681, −4.932649172765633, −4.573282588876985, −3.644567249663952, −3.280274328812413, −2.293734978683200, −1.781256438924233, −0.7917172770481941, 0,
0.7917172770481941, 1.781256438924233, 2.293734978683200, 3.280274328812413, 3.644567249663952, 4.573282588876985, 4.932649172765633, 5.718474132690681, 6.045056732645038, 6.540215605882846, 7.377872702278671, 7.853156987653098, 8.247093816047277, 8.964884723188614, 9.397435342591218, 10.24241899311301, 10.59718188851624, 11.02908923178337, 11.28875544921153, 12.13165520614327, 12.68338608410251, 13.06734600125655, 13.41514023970356, 13.94727152609605, 14.64212896273003
