L(s) = 1 | − 6·13-s + 8·17-s − 4·29-s − 2·37-s + 8·41-s − 7·49-s + 4·53-s + 10·61-s − 6·73-s − 16·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.66·13-s + 1.94·17-s − 0.742·29-s − 0.328·37-s + 1.24·41-s − 49-s + 0.549·53-s + 1.28·61-s − 0.702·73-s − 1.69·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.781996718\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.781996718\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.26270648035631, −15.50999581467826, −14.86920729152731, −14.35226644160998, −14.22286082238002, −13.16215763577493, −12.73078851081897, −12.13073873704581, −11.77006813113095, −11.00964528298145, −10.29874397864088, −9.740958698420582, −9.485134824932497, −8.546499042769316, −7.866159085841111, −7.403620027755322, −6.896370513678137, −5.887131394337717, −5.431390921798599, −4.797920452177807, −3.991657332270240, −3.203305498468879, −2.535629709683144, −1.636959756070308, −0.5879881805080925,
0.5879881805080925, 1.636959756070308, 2.535629709683144, 3.203305498468879, 3.991657332270240, 4.797920452177807, 5.431390921798599, 5.887131394337717, 6.896370513678137, 7.403620027755322, 7.866159085841111, 8.546499042769316, 9.485134824932497, 9.740958698420582, 10.29874397864088, 11.00964528298145, 11.77006813113095, 12.13073873704581, 12.73078851081897, 13.16215763577493, 14.22286082238002, 14.35226644160998, 14.86920729152731, 15.50999581467826, 16.26270648035631