| L(s) = 1 | − 3-s + 5-s + 3·7-s + 9-s − 3·11-s − 15-s − 17-s + 5·19-s − 3·21-s + 25-s − 27-s − 5·29-s − 4·31-s + 3·33-s + 3·35-s − 9·37-s − 7·41-s − 8·43-s + 45-s + 3·47-s + 2·49-s + 51-s − 7·53-s − 3·55-s − 5·57-s − 6·59-s − 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.258·15-s − 0.242·17-s + 1.14·19-s − 0.654·21-s + 1/5·25-s − 0.192·27-s − 0.928·29-s − 0.718·31-s + 0.522·33-s + 0.507·35-s − 1.47·37-s − 1.09·41-s − 1.21·43-s + 0.149·45-s + 0.437·47-s + 2/7·49-s + 0.140·51-s − 0.961·53-s − 0.404·55-s − 0.662·57-s − 0.781·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.76746941341311, −12.22597094643054, −11.91368608734279, −11.31823602666375, −10.98043460257010, −10.68552849618581, −10.09846425270742, −9.681632787157477, −9.210407167955784, −8.633183730818132, −8.153426738636823, −7.666683205930380, −7.366073156776606, −6.672059933857252, −6.350933238223119, −5.486363518161623, −5.252701676432209, −5.041769063896530, −4.472357837798281, −3.550177261562472, −3.384325900157101, −2.455570493206232, −1.821960723207328, −1.614911254866248, −0.7353572107480194, 0,
0.7353572107480194, 1.614911254866248, 1.821960723207328, 2.455570493206232, 3.384325900157101, 3.550177261562472, 4.472357837798281, 5.041769063896530, 5.252701676432209, 5.486363518161623, 6.350933238223119, 6.672059933857252, 7.366073156776606, 7.666683205930380, 8.153426738636823, 8.633183730818132, 9.210407167955784, 9.681632787157477, 10.09846425270742, 10.68552849618581, 10.98043460257010, 11.31823602666375, 11.91368608734279, 12.22597094643054, 12.76746941341311
