L(s) = 1 | − 2·5-s + 7-s − 3·9-s + 4·13-s − 4·17-s − 2·19-s + 23-s − 25-s + 6·29-s − 2·31-s − 2·35-s + 6·37-s − 2·41-s + 4·43-s + 6·45-s − 6·47-s + 49-s − 6·53-s + 2·61-s − 3·63-s − 8·65-s − 4·67-s − 12·71-s − 10·73-s + 8·79-s + 9·81-s − 6·83-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s + 1.10·13-s − 0.970·17-s − 0.458·19-s + 0.208·23-s − 1/5·25-s + 1.11·29-s − 0.359·31-s − 0.338·35-s + 0.986·37-s − 0.312·41-s + 0.609·43-s + 0.894·45-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.256·61-s − 0.377·63-s − 0.992·65-s − 0.488·67-s − 1.42·71-s − 1.17·73-s + 0.900·79-s + 81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.207033876\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.207033876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−16.50563079328396, −15.93724560006153, −15.56473276261074, −14.82209283632071, −14.46222841013482, −13.67513110655361, −13.23387218825541, −12.52427200379896, −11.70244087455485, −11.46508576473854, −10.89660389194811, −10.37423125629245, −9.320016651264260, −8.769393749005160, −8.263240837173680, −7.824253477771573, −6.935993202393052, −6.246723734090923, −5.713947314832887, −4.700222414335502, −4.235657304180335, −3.398621414589620, −2.708594169677382, −1.682170454284921, −0.5131196689022875,
0.5131196689022875, 1.682170454284921, 2.708594169677382, 3.398621414589620, 4.235657304180335, 4.700222414335502, 5.713947314832887, 6.246723734090923, 6.935993202393052, 7.824253477771573, 8.263240837173680, 8.769393749005160, 9.320016651264260, 10.37423125629245, 10.89660389194811, 11.46508576473854, 11.70244087455485, 12.52427200379896, 13.23387218825541, 13.67513110655361, 14.46222841013482, 14.82209283632071, 15.56473276261074, 15.93724560006153, 16.50563079328396