| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 6·13-s + 15-s + 2·17-s + 25-s − 27-s + 2·29-s + 4·31-s + 4·33-s + 10·37-s − 6·39-s − 6·41-s + 43-s − 45-s − 7·49-s − 2·51-s + 6·53-s + 4·55-s − 4·59-s + 2·61-s − 6·65-s + 12·67-s − 16·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s + 0.485·17-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.696·33-s + 1.64·37-s − 0.960·39-s − 0.937·41-s + 0.152·43-s − 0.149·45-s − 49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.744·65-s + 1.46·67-s − 1.89·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.729852999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.729852999\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.94004622543428, −14.10724735161778, −13.67459532945855, −12.97697189840720, −12.88669667257306, −12.07892990210623, −11.44159625947946, −11.27190691576169, −10.48835151838970, −10.26476968907306, −9.555798809708180, −8.816631587423271, −8.213327231046950, −7.913554962725056, −7.264692532794304, −6.462112895929333, −6.127661922312156, −5.455670800454409, −4.888044432548862, −4.271249996549301, −3.557330695860771, −3.009480275616221, −2.157666797054068, −1.182785066598597, −0.5634998962282630,
0.5634998962282630, 1.182785066598597, 2.157666797054068, 3.009480275616221, 3.557330695860771, 4.271249996549301, 4.888044432548862, 5.455670800454409, 6.127661922312156, 6.462112895929333, 7.264692532794304, 7.913554962725056, 8.213327231046950, 8.816631587423271, 9.555798809708180, 10.26476968907306, 10.48835151838970, 11.27190691576169, 11.44159625947946, 12.07892990210623, 12.88669667257306, 12.97697189840720, 13.67459532945855, 14.10724735161778, 14.94004622543428
