L(s) = 1 | + 2·3-s + 5-s + 3·7-s + 9-s + 5·11-s + 4·13-s + 2·15-s − 3·17-s − 19-s + 6·21-s − 8·23-s − 4·25-s − 4·27-s + 2·29-s − 4·31-s + 10·33-s + 3·35-s − 10·37-s + 8·39-s + 10·41-s + 43-s + 45-s + 47-s + 2·49-s − 6·51-s + 4·53-s + 5·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-s + 0.516·15-s − 0.727·17-s − 0.229·19-s + 1.30·21-s − 1.66·23-s − 4/5·25-s − 0.769·27-s + 0.371·29-s − 0.718·31-s + 1.74·33-s + 0.507·35-s − 1.64·37-s + 1.28·39-s + 1.56·41-s + 0.152·43-s + 0.149·45-s + 0.145·47-s + 2/7·49-s − 0.840·51-s + 0.549·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.076206070\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.076206070\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.470845631762160929353564163989, −8.762972910391102183787591721206, −8.381087888928782814621554216261, −7.46802549489449929304458830547, −6.39486340421971046323119110374, −5.61135260065455155907566257543, −4.17947794613570531538770424508, −3.73571330395388729025783799541, −2.22354879090908877160738042469, −1.55214010532178794455093749689,
1.55214010532178794455093749689, 2.22354879090908877160738042469, 3.73571330395388729025783799541, 4.17947794613570531538770424508, 5.61135260065455155907566257543, 6.39486340421971046323119110374, 7.46802549489449929304458830547, 8.381087888928782814621554216261, 8.762972910391102183787591721206, 9.470845631762160929353564163989