L(s) = 1 | + 2-s + 4-s − 2·5-s + 4·7-s + 8-s − 2·10-s + 6·13-s + 4·14-s + 16-s + 2·17-s − 4·19-s − 2·20-s − 4·23-s − 25-s + 6·26-s + 4·28-s + 6·29-s + 32-s + 2·34-s − 8·35-s + 6·37-s − 4·38-s − 2·40-s − 6·41-s − 4·43-s − 4·46-s + 12·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s + 0.353·8-s − 0.632·10-s + 1.66·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.447·20-s − 0.834·23-s − 1/5·25-s + 1.17·26-s + 0.755·28-s + 1.11·29-s + 0.176·32-s + 0.342·34-s − 1.35·35-s + 0.986·37-s − 0.648·38-s − 0.316·40-s − 0.937·41-s − 0.609·43-s − 0.589·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.069873472\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.069873472\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.645953380304319851959090735294, −8.212806450751299664087531727275, −7.68830677207914579917009283814, −6.60103962765417584469770170019, −5.84860418329322714481021224574, −4.91374385626968641904300752674, −4.14534836475784804647223271983, −3.60112495689583876906449580237, −2.22344095936308660753676570881, −1.13252217573474415409866716509,
1.13252217573474415409866716509, 2.22344095936308660753676570881, 3.60112495689583876906449580237, 4.14534836475784804647223271983, 4.91374385626968641904300752674, 5.84860418329322714481021224574, 6.60103962765417584469770170019, 7.68830677207914579917009283814, 8.212806450751299664087531727275, 8.645953380304319851959090735294