L(s) = 1 | + 4·13-s − 8·17-s − 10·29-s + 12·37-s + 10·41-s − 7·49-s + 4·53-s + 10·61-s + 16·73-s + 10·89-s − 8·97-s + 2·101-s + 6·109-s + 16·113-s + ⋯ |
L(s) = 1 | + 1.10·13-s − 1.94·17-s − 1.85·29-s + 1.97·37-s + 1.56·41-s − 49-s + 0.549·53-s + 1.28·61-s + 1.87·73-s + 1.05·89-s − 0.812·97-s + 0.199·101-s + 0.574·109-s + 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.810981779\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.810981779\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.935384661251128756487087682079, −7.23992932349289750072857579645, −6.40341818141713182323725618147, −6.00178162513688601397902655350, −5.08483969514461699201315648855, −4.20653677488776657079886207199, −3.74502694198768108207542352534, −2.60329712062717383983860995277, −1.88203915631401830676702420061, −0.67503075954131059935836478299,
0.67503075954131059935836478299, 1.88203915631401830676702420061, 2.60329712062717383983860995277, 3.74502694198768108207542352534, 4.20653677488776657079886207199, 5.08483969514461699201315648855, 6.00178162513688601397902655350, 6.40341818141713182323725618147, 7.23992932349289750072857579645, 7.935384661251128756487087682079