L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 15-s − 17-s − 4·19-s + 2·21-s + 25-s + 27-s + 6·29-s + 10·31-s − 2·35-s + 4·37-s + 2·41-s + 4·43-s − 45-s + 6·47-s − 3·49-s − 51-s − 6·53-s − 4·57-s + 6·59-s + 6·61-s + 2·63-s + 8·67-s − 12·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.258·15-s − 0.242·17-s − 0.917·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s − 0.338·35-s + 0.657·37-s + 0.312·41-s + 0.609·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.140·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s + 0.768·61-s + 0.251·63-s + 0.977·67-s − 1.42·71-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
\(L(1)\) |
\(\approx\) |
\(4.213410847\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.213410847\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.63731145515804, −11.99392315568743, −11.77069231284247, −11.16468235211402, −10.76022405768178, −10.35187618936084, −9.816339946360426, −9.336036081866783, −8.790250360615516, −8.349793971199526, −8.000809668576948, −7.768219936586688, −6.929957804213555, −6.649278502087080, −6.143498900149717, −5.456283592409191, −4.865043140099968, −4.386935596290625, −4.160898504394949, −3.459662111022784, −2.766246449972634, −2.438381277613744, −1.816704753505812, −1.042746879155686, −0.5798556985903863,
0.5798556985903863, 1.042746879155686, 1.816704753505812, 2.438381277613744, 2.766246449972634, 3.459662111022784, 4.160898504394949, 4.386935596290625, 4.865043140099968, 5.456283592409191, 6.143498900149717, 6.649278502087080, 6.929957804213555, 7.768219936586688, 8.000809668576948, 8.349793971199526, 8.790250360615516, 9.336036081866783, 9.816339946360426, 10.35187618936084, 10.76022405768178, 11.16468235211402, 11.77069231284247, 11.99392315568743, 12.63731145515804