| L(s) = 1 | + 27·3-s + 832·7-s + 729·9-s − 2.48e3·11-s − 1.48e4·13-s + 2.23e4·17-s − 1.63e4·19-s + 2.24e4·21-s + 1.15e5·23-s + 1.96e4·27-s + 1.57e5·29-s − 1.64e4·31-s − 6.70e4·33-s + 1.49e5·37-s − 4.01e5·39-s − 2.41e5·41-s + 4.43e5·43-s − 9.22e5·47-s − 1.31e5·49-s + 6.02e5·51-s + 6.97e5·53-s − 4.40e5·57-s + 8.70e5·59-s + 2.06e6·61-s + 6.06e5·63-s + 1.68e6·67-s + 3.10e6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.916·7-s + 1/3·9-s − 0.562·11-s − 1.87·13-s + 1.10·17-s − 0.545·19-s + 0.529·21-s + 1.97·23-s + 0.192·27-s + 1.19·29-s − 0.0992·31-s − 0.324·33-s + 0.484·37-s − 1.08·39-s − 0.546·41-s + 0.850·43-s − 1.29·47-s − 0.159·49-s + 0.635·51-s + 0.643·53-s − 0.314·57-s + 0.551·59-s + 1.16·61-s + 0.305·63-s + 0.682·67-s + 1.13·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.926295580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.926295580\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 832 T + p^{7} T^{2} \) |
| 11 | \( 1 + 2484 T + p^{7} T^{2} \) |
| 13 | \( 1 + 14870 T + p^{7} T^{2} \) |
| 17 | \( 1 - 22302 T + p^{7} T^{2} \) |
| 19 | \( 1 + 16300 T + p^{7} T^{2} \) |
| 23 | \( 1 - 115128 T + p^{7} T^{2} \) |
| 29 | \( 1 - 157086 T + p^{7} T^{2} \) |
| 31 | \( 1 + 16456 T + p^{7} T^{2} \) |
| 37 | \( 1 - 149266 T + p^{7} T^{2} \) |
| 41 | \( 1 + 241110 T + p^{7} T^{2} \) |
| 43 | \( 1 - 443188 T + p^{7} T^{2} \) |
| 47 | \( 1 + 922752 T + p^{7} T^{2} \) |
| 53 | \( 1 - 697626 T + p^{7} T^{2} \) |
| 59 | \( 1 - 870156 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2067062 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1680748 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1070280 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2403334 T + p^{7} T^{2} \) |
| 79 | \( 1 - 2301512 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4708692 T + p^{7} T^{2} \) |
| 89 | \( 1 - 4143690 T + p^{7} T^{2} \) |
| 97 | \( 1 - 1622974 T + p^{7} T^{2} \) |
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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
−10.38679206126442346188162805833, −9.628845802834803125060813091625, −8.501594077325419952471121244826, −7.71933196636968553432169328155, −6.92383772701490388155190618111, −5.24320718126269361881268894199, −4.65237896635824375931403486902, −3.08269352211828996113368923534, −2.17626255089911316032973593356, −0.821583130773418500157004000766,
0.821583130773418500157004000766, 2.17626255089911316032973593356, 3.08269352211828996113368923534, 4.65237896635824375931403486902, 5.24320718126269361881268894199, 6.92383772701490388155190618111, 7.71933196636968553432169328155, 8.501594077325419952471121244826, 9.628845802834803125060813091625, 10.38679206126442346188162805833
