| L(s) = 1 | + 82·3-s − 448·5-s − 343·7-s + 4.53e3·9-s − 2.40e3·11-s − 7.11e3·13-s − 3.67e4·15-s + 2.48e3·17-s − 3.64e4·19-s − 2.81e4·21-s − 1.28e4·23-s + 1.22e5·25-s + 1.92e5·27-s + 8.80e4·29-s + 2.82e5·31-s − 1.97e5·33-s + 1.53e5·35-s + 2.14e5·37-s − 5.83e5·39-s − 1.40e5·41-s − 3.64e4·43-s − 2.03e6·45-s + 7.16e5·47-s + 1.17e5·49-s + 2.03e5·51-s + 5.69e4·53-s + 1.07e6·55-s + ⋯ |
| L(s) = 1 | + 1.75·3-s − 1.60·5-s − 0.377·7-s + 2.07·9-s − 0.545·11-s − 0.898·13-s − 2.81·15-s + 0.122·17-s − 1.22·19-s − 0.662·21-s − 0.220·23-s + 1.56·25-s + 1.88·27-s + 0.670·29-s + 1.70·31-s − 0.956·33-s + 0.605·35-s + 0.696·37-s − 1.57·39-s − 0.319·41-s − 0.0699·43-s − 3.32·45-s + 1.00·47-s + 1/7·49-s + 0.215·51-s + 0.0525·53-s + 0.874·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.427850747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427850747\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 - 82 T + p^{7} T^{2} \) |
| 5 | \( 1 + 448 T + p^{7} T^{2} \) |
| 11 | \( 1 + 2408 T + p^{7} T^{2} \) |
| 13 | \( 1 + 7116 T + p^{7} T^{2} \) |
| 17 | \( 1 - 2486 T + p^{7} T^{2} \) |
| 19 | \( 1 + 36482 T + p^{7} T^{2} \) |
| 23 | \( 1 + 560 p T + p^{7} T^{2} \) |
| 29 | \( 1 - 88094 T + p^{7} T^{2} \) |
| 31 | \( 1 - 282636 T + p^{7} T^{2} \) |
| 37 | \( 1 - 214534 T + p^{7} T^{2} \) |
| 41 | \( 1 + 140874 T + p^{7} T^{2} \) |
| 43 | \( 1 + 848 p T + p^{7} T^{2} \) |
| 47 | \( 1 - 716868 T + p^{7} T^{2} \) |
| 53 | \( 1 - 56946 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2149862 T + p^{7} T^{2} \) |
| 61 | \( 1 + 3084360 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3034364 T + p^{7} T^{2} \) |
| 71 | \( 1 + 106624 T + p^{7} T^{2} \) |
| 73 | \( 1 - 988930 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3415896 T + p^{7} T^{2} \) |
| 83 | \( 1 - 15142 T + p^{7} T^{2} \) |
| 89 | \( 1 - 174810 T + p^{7} T^{2} \) |
| 97 | \( 1 - 13506790 T + p^{7} T^{2} \) |
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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
−9.806207674552223354668783308294, −8.749388504223389539695728101465, −8.109324763074551369999524728682, −7.58459764726094128454516346495, −6.64779982613483135903860915660, −4.66675749273849355766736938074, −3.98611642291487955410316333671, −3.02890943762143508508021789781, −2.31252368451662216208514788435, −0.61726263358203667645520402402,
0.61726263358203667645520402402, 2.31252368451662216208514788435, 3.02890943762143508508021789781, 3.98611642291487955410316333671, 4.66675749273849355766736938074, 6.64779982613483135903860915660, 7.58459764726094128454516346495, 8.109324763074551369999524728682, 8.749388504223389539695728101465, 9.806207674552223354668783308294
