| L(s) = 1 | − 5-s + 2·11-s − 13-s + 17-s − 6·19-s − 4·25-s − 3·29-s + 5·31-s − 4·37-s + 9·41-s − 6·43-s + 7·47-s + 10·53-s − 2·55-s + 9·59-s − 6·61-s + 65-s + 8·67-s − 2·71-s − 10·73-s − 17·83-s − 85-s − 4·89-s + 6·95-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s − 0.277·13-s + 0.242·17-s − 1.37·19-s − 4/5·25-s − 0.557·29-s + 0.898·31-s − 0.657·37-s + 1.40·41-s − 0.914·43-s + 1.02·47-s + 1.37·53-s − 0.269·55-s + 1.17·59-s − 0.768·61-s + 0.124·65-s + 0.977·67-s − 0.237·71-s − 1.17·73-s − 1.86·83-s − 0.108·85-s − 0.423·89-s + 0.615·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.80741758153861, −13.26905730342687, −12.86349597818416, −12.24623661525902, −11.94321236645786, −11.40914866894178, −11.00776193260394, −10.28615374232575, −10.05698997038061, −9.392916747158932, −8.787520673113261, −8.499321984669329, −7.890864032381670, −7.320824758116661, −6.933834605415948, −6.274862958869618, −5.830932635605256, −5.255866188208058, −4.478061892999306, −4.083144909964665, −3.687506634120012, −2.839428545186949, −2.276065041589698, −1.619846529531606, −0.7874183153889334, 0,
0.7874183153889334, 1.619846529531606, 2.276065041589698, 2.839428545186949, 3.687506634120012, 4.083144909964665, 4.478061892999306, 5.255866188208058, 5.830932635605256, 6.274862958869618, 6.933834605415948, 7.320824758116661, 7.890864032381670, 8.499321984669329, 8.787520673113261, 9.392916747158932, 10.05698997038061, 10.28615374232575, 11.00776193260394, 11.40914866894178, 11.94321236645786, 12.24623661525902, 12.86349597818416, 13.26905730342687, 13.80741758153861
