| L(s) = 1 | − 3-s + 9-s + 5·11-s + 13-s − 2·17-s + 7·19-s + 3·23-s − 27-s − 6·31-s − 5·33-s + 5·37-s − 39-s + 9·41-s + 10·43-s + 13·47-s + 2·51-s + 53-s − 7·57-s + 4·59-s + 2·61-s + 6·67-s − 3·69-s + 2·71-s + 4·73-s + 14·79-s + 81-s + 10·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.485·17-s + 1.60·19-s + 0.625·23-s − 0.192·27-s − 1.07·31-s − 0.870·33-s + 0.821·37-s − 0.160·39-s + 1.40·41-s + 1.52·43-s + 1.89·47-s + 0.280·51-s + 0.137·53-s − 0.927·57-s + 0.520·59-s + 0.256·61-s + 0.733·67-s − 0.361·69-s + 0.237·71-s + 0.468·73-s + 1.57·79-s + 1/9·81-s + 1.09·83-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
| \(L(1)\) |
\(\approx\) |
\(3.077919203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.077919203\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.26332438636786, −13.91809607330338, −13.33943298000872, −12.66144457670162, −12.32196691369626, −11.74532158570147, −11.24846199047695, −10.96700270274880, −10.34973272802419, −9.474915322654702, −9.260752858238102, −8.946629534318281, −7.936623231258914, −7.505377024349614, −6.897785094235104, −6.505836739811693, −5.702126892921779, −5.511108424719917, −4.633692694786946, −4.012802590584426, −3.666111001106252, −2.756051354102955, −2.028432497860164, −1.016954171216186, −0.8159488504394748,
0.8159488504394748, 1.016954171216186, 2.028432497860164, 2.756051354102955, 3.666111001106252, 4.012802590584426, 4.633692694786946, 5.511108424719917, 5.702126892921779, 6.505836739811693, 6.897785094235104, 7.505377024349614, 7.936623231258914, 8.946629534318281, 9.260752858238102, 9.474915322654702, 10.34973272802419, 10.96700270274880, 11.24846199047695, 11.74532158570147, 12.32196691369626, 12.66144457670162, 13.33943298000872, 13.91809607330338, 14.26332438636786
