L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 3·11-s − 3·13-s + 15-s + 17-s − 6·19-s + 21-s + 2·23-s − 4·25-s + 27-s + 6·29-s − 4·31-s − 3·33-s + 35-s − 11·37-s − 3·39-s + 12·41-s − 3·43-s + 45-s − 12·47-s + 49-s + 51-s + 5·53-s − 3·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.832·13-s + 0.258·15-s + 0.242·17-s − 1.37·19-s + 0.218·21-s + 0.417·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s − 1.80·37-s − 0.480·39-s + 1.87·41-s − 0.457·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.140·51-s + 0.686·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.87710609360113937547424237321, −7.15846000851807483821264157910, −6.40847598971496038123703884301, −5.51380757839723608370307794849, −4.86661971826858811827994234789, −4.12668066565654916480762639371, −3.06417803097324762901463639854, −2.34967609752030194474570843769, −1.60547260706163950648585976883, 0,
1.60547260706163950648585976883, 2.34967609752030194474570843769, 3.06417803097324762901463639854, 4.12668066565654916480762639371, 4.86661971826858811827994234789, 5.51380757839723608370307794849, 6.40847598971496038123703884301, 7.15846000851807483821264157910, 7.87710609360113937547424237321