L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s − 11-s − 12-s + 4·13-s − 4·14-s + 16-s + 17-s − 18-s − 8·19-s − 4·21-s + 22-s + 24-s − 4·26-s − 27-s + 4·28-s + 10·31-s − 32-s + 33-s − 34-s + 36-s − 8·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1.10·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.872·21-s + 0.213·22-s + 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.755·28-s + 1.79·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.438412939\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.438412939\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.45261383035368, −14.82842890777466, −14.23773216337742, −13.58563115475307, −13.13374120747918, −12.27604365638443, −11.91464773787323, −11.33530289267552, −10.88315857108563, −10.38954853473780, −10.10676793941745, −9.020237647324648, −8.478578589515048, −8.307090417113622, −7.611837626986431, −6.901317657458619, −6.301091253164449, −5.806367272090629, −4.992669162984812, −4.527578871916769, −3.817924298347368, −2.849427365587474, −1.892210298365421, −1.496918316780178, −0.5581839109961046,
0.5581839109961046, 1.496918316780178, 1.892210298365421, 2.849427365587474, 3.817924298347368, 4.527578871916769, 4.992669162984812, 5.806367272090629, 6.301091253164449, 6.901317657458619, 7.611837626986431, 8.307090417113622, 8.478578589515048, 9.020237647324648, 10.10676793941745, 10.38954853473780, 10.88315857108563, 11.33530289267552, 11.91464773787323, 12.27604365638443, 13.13374120747918, 13.58563115475307, 14.23773216337742, 14.82842890777466, 15.45261383035368