| L(s) = 1 | − 3-s − 4·7-s + 9-s − 2·11-s + 13-s − 4·17-s − 2·19-s + 4·21-s + 6·23-s − 27-s + 2·29-s − 4·31-s + 2·33-s − 6·37-s − 39-s − 6·41-s + 8·43-s − 8·47-s + 9·49-s + 4·51-s + 10·53-s + 2·57-s + 14·59-s − 10·61-s − 4·63-s + 4·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.970·17-s − 0.458·19-s + 0.872·21-s + 1.25·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 9/7·49-s + 0.560·51-s + 1.37·53-s + 0.264·57-s + 1.82·59-s − 1.28·61-s − 0.503·63-s + 0.488·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.65775640458320, −13.74528134836115, −13.41165763370987, −12.95289800149272, −12.62655353219589, −12.11458091527693, −11.37564169446532, −10.97467896540117, −10.44737351767235, −10.02033792910855, −9.480605412890657, −8.787456997101811, −8.601187029844751, −7.634992663164785, −6.961094954973343, −6.750076868389122, −6.203367509739489, −5.501103686522403, −5.122542770628501, −4.300351215014197, −3.758473607663157, −3.067080286277707, −2.538398752756803, −1.694174365995474, −0.6691563813121786, 0,
0.6691563813121786, 1.694174365995474, 2.538398752756803, 3.067080286277707, 3.758473607663157, 4.300351215014197, 5.122542770628501, 5.501103686522403, 6.203367509739489, 6.750076868389122, 6.961094954973343, 7.634992663164785, 8.601187029844751, 8.787456997101811, 9.480605412890657, 10.02033792910855, 10.44737351767235, 10.97467896540117, 11.37564169446532, 12.11458091527693, 12.62655353219589, 12.95289800149272, 13.41165763370987, 13.74528134836115, 14.65775640458320
