| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 3·11-s − 15-s − 6·17-s + 19-s − 21-s + 23-s + 25-s + 27-s − 2·29-s − 4·31-s − 3·33-s + 35-s + 2·37-s + 5·41-s + 6·43-s − 45-s + 47-s + 49-s − 6·51-s + 13·53-s + 3·55-s + 57-s − 3·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.258·15-s − 1.45·17-s + 0.229·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s + 0.328·37-s + 0.780·41-s + 0.914·43-s − 0.149·45-s + 0.145·47-s + 1/7·49-s − 0.840·51-s + 1.78·53-s + 0.404·55-s + 0.132·57-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−15.13140739104205, −14.73097189531742, −13.84283977270895, −13.62290899165975, −13.00317498022006, −12.60219731765013, −12.08487607274576, −11.28643701873973, −10.80602578184387, −10.53912306623761, −9.520373933629048, −9.354353313916351, −8.682371537176916, −8.107437595357978, −7.614824949799624, −7.041459688217238, −6.559627464322477, −5.724490004338992, −5.188158871888930, −4.376738154848260, −3.962335576334222, −3.213638707132621, −2.515278853388106, −2.086003528783661, −0.9045181332078536, 0,
0.9045181332078536, 2.086003528783661, 2.515278853388106, 3.213638707132621, 3.962335576334222, 4.376738154848260, 5.188158871888930, 5.724490004338992, 6.559627464322477, 7.041459688217238, 7.614824949799624, 8.107437595357978, 8.682371537176916, 9.354353313916351, 9.520373933629048, 10.53912306623761, 10.80602578184387, 11.28643701873973, 12.08487607274576, 12.60219731765013, 13.00317498022006, 13.62290899165975, 13.84283977270895, 14.73097189531742, 15.13140739104205
