L(s) = 1 | − 2·5-s + 7-s − 2·11-s − 13-s + 6·17-s + 5·19-s − 6·23-s − 25-s − 8·29-s + 8·31-s − 2·35-s + 5·37-s + 8·41-s + 4·43-s + 10·47-s − 6·49-s − 4·53-s + 4·55-s + 14·59-s − 3·61-s + 2·65-s + 13·67-s + 4·71-s + 9·73-s − 2·77-s + 11·79-s − 12·83-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.603·11-s − 0.277·13-s + 1.45·17-s + 1.14·19-s − 1.25·23-s − 1/5·25-s − 1.48·29-s + 1.43·31-s − 0.338·35-s + 0.821·37-s + 1.24·41-s + 0.609·43-s + 1.45·47-s − 6/7·49-s − 0.549·53-s + 0.539·55-s + 1.82·59-s − 0.384·61-s + 0.248·65-s + 1.58·67-s + 0.474·71-s + 1.05·73-s − 0.227·77-s + 1.23·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.417891343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.417891343\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−9.517026791789933025951612270668, −8.141441334825677566937789349991, −7.86749870783756385045860969127, −7.24764130899628727569423119197, −5.92718935464768987920858473492, −5.29554848587607308034135507932, −4.23592786371004028357777676518, −3.47453539619149025125996376504, −2.35447010108907921399113958644, −0.826140575077611309933401896495,
0.826140575077611309933401896495, 2.35447010108907921399113958644, 3.47453539619149025125996376504, 4.23592786371004028357777676518, 5.29554848587607308034135507932, 5.92718935464768987920858473492, 7.24764130899628727569423119197, 7.86749870783756385045860969127, 8.141441334825677566937789349991, 9.517026791789933025951612270668