| L(s) = 1 | − 2·3-s − 3·5-s + 4·7-s + 9-s + 6·13-s + 6·15-s + 3·17-s − 19-s − 8·21-s − 4·23-s + 4·25-s + 4·27-s + 8·29-s − 5·31-s − 12·35-s − 8·37-s − 12·39-s − 12·41-s − 7·43-s − 3·45-s + 2·47-s + 9·49-s − 6·51-s − 11·53-s + 2·57-s + 4·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 1.51·7-s + 1/3·9-s + 1.66·13-s + 1.54·15-s + 0.727·17-s − 0.229·19-s − 1.74·21-s − 0.834·23-s + 4/5·25-s + 0.769·27-s + 1.48·29-s − 0.898·31-s − 2.02·35-s − 1.31·37-s − 1.92·39-s − 1.87·41-s − 1.06·43-s − 0.447·45-s + 0.291·47-s + 9/7·49-s − 0.840·51-s − 1.51·53-s + 0.264·57-s + 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160204 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 331 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.64533801991451, −12.86811073235119, −12.23737311324345, −11.87119774742213, −11.73962919605663, −11.19332600100285, −10.82045378466501, −10.47249747813362, −9.961019272973811, −8.936295988721571, −8.504353030495415, −8.124701915582213, −7.962330271490891, −7.072252518701747, −6.720655360921421, −6.110543279389474, −5.514466300871376, −5.112250830967945, −4.607149578532601, −4.092718911511573, −3.514930725161716, −3.075087375502376, −1.825114967555702, −1.461403417839792, −0.7302083922643062, 0,
0.7302083922643062, 1.461403417839792, 1.825114967555702, 3.075087375502376, 3.514930725161716, 4.092718911511573, 4.607149578532601, 5.112250830967945, 5.514466300871376, 6.110543279389474, 6.720655360921421, 7.072252518701747, 7.962330271490891, 8.124701915582213, 8.504353030495415, 8.936295988721571, 9.961019272973811, 10.47249747813362, 10.82045378466501, 11.19332600100285, 11.73962919605663, 11.87119774742213, 12.23737311324345, 12.86811073235119, 13.64533801991451
