| L(s) = 1 | − 27·3-s − 410·5-s + 1.02e3·7-s + 729·9-s + 1.33e3·11-s + 1.29e4·13-s + 1.10e4·15-s + 1.70e4·17-s + 5.41e4·19-s − 2.77e4·21-s + 1.14e4·23-s + 8.99e4·25-s − 1.96e4·27-s − 1.86e5·29-s + 1.88e5·31-s − 3.59e4·33-s − 4.21e5·35-s + 3.95e5·37-s − 3.49e5·39-s − 4.75e4·41-s − 6.02e5·43-s − 2.98e5·45-s + 6.47e5·47-s + 2.33e5·49-s − 4.60e5·51-s − 1.31e6·53-s − 5.45e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.46·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 1.63·13-s + 0.846·15-s + 0.842·17-s + 1.81·19-s − 0.654·21-s + 0.196·23-s + 1.15·25-s − 0.192·27-s − 1.42·29-s + 1.13·31-s − 0.174·33-s − 1.66·35-s + 1.28·37-s − 0.944·39-s − 0.107·41-s − 1.15·43-s − 0.488·45-s + 0.909·47-s + 0.283·49-s − 0.486·51-s − 1.21·53-s − 0.442·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.091396415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091396415\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 11 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 82 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 1028 T + p^{7} T^{2} \) |
| 13 | \( 1 - 12958 T + p^{7} T^{2} \) |
| 17 | \( 1 - 17062 T + p^{7} T^{2} \) |
| 19 | \( 1 - 54168 T + p^{7} T^{2} \) |
| 23 | \( 1 - 11488 T + p^{7} T^{2} \) |
| 29 | \( 1 + 186654 T + p^{7} T^{2} \) |
| 31 | \( 1 - 188672 T + p^{7} T^{2} \) |
| 37 | \( 1 - 395886 T + p^{7} T^{2} \) |
| 41 | \( 1 + 47546 T + p^{7} T^{2} \) |
| 43 | \( 1 + 602088 T + p^{7} T^{2} \) |
| 47 | \( 1 - 647200 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1312722 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2681140 T + p^{7} T^{2} \) |
| 61 | \( 1 - 551190 T + p^{7} T^{2} \) |
| 67 | \( 1 + 459260 T + p^{7} T^{2} \) |
| 71 | \( 1 - 18072 T + p^{7} T^{2} \) |
| 73 | \( 1 + 426062 T + p^{7} T^{2} \) |
| 79 | \( 1 + 297764 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5684028 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6342966 T + p^{7} T^{2} \) |
| 97 | \( 1 - 16651586 T + p^{7} T^{2} \) |
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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
−9.780289271788683821494120990911, −8.558082437740715769120359431367, −7.88679514690283763162928700376, −7.23239936424328221424302536498, −5.95093869394717217914711902472, −5.02384883181829298952184809546, −4.04031829721025046353634687962, −3.29768065615376745019709338995, −1.38303212758836796694098902324, −0.75540283288340822806579044615,
0.75540283288340822806579044615, 1.38303212758836796694098902324, 3.29768065615376745019709338995, 4.04031829721025046353634687962, 5.02384883181829298952184809546, 5.95093869394717217914711902472, 7.23239936424328221424302536498, 7.88679514690283763162928700376, 8.558082437740715769120359431367, 9.780289271788683821494120990911
