L(s) = 1 | − 2·13-s + 6·17-s + 4·19-s − 8·23-s + 2·29-s − 4·31-s − 10·37-s − 2·41-s − 4·43-s − 8·47-s − 7·49-s − 2·53-s + 8·59-s − 2·61-s − 12·67-s + 8·71-s + 14·73-s + 12·79-s + 4·83-s + 14·89-s − 2·97-s + 10·101-s + 8·103-s − 12·107-s − 10·109-s − 10·113-s + ⋯ |
L(s) = 1 | − 0.554·13-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 0.371·29-s − 0.718·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s − 49-s − 0.274·53-s + 1.04·59-s − 0.256·61-s − 1.46·67-s + 0.949·71-s + 1.63·73-s + 1.35·79-s + 0.439·83-s + 1.48·89-s − 0.203·97-s + 0.995·101-s + 0.788·103-s − 1.16·107-s − 0.957·109-s − 0.940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.84576753749159553189051136542, −6.86673020774443852349642118582, −6.21571589573351057050786162267, −5.28798849739880521329879889241, −4.98632065352668771612110993791, −3.72252159622727809194264912287, −3.33775017663440405459599489936, −2.20506432534604625006328712908, −1.33562232876556715974158588890, 0,
1.33562232876556715974158588890, 2.20506432534604625006328712908, 3.33775017663440405459599489936, 3.72252159622727809194264912287, 4.98632065352668771612110993791, 5.28798849739880521329879889241, 6.21571589573351057050786162267, 6.86673020774443852349642118582, 7.84576753749159553189051136542