| L(s) = 1 | − 102·3-s + 6.18e3·7-s + 3.84e3·9-s + 1.45e4·13-s − 1.60e5·19-s − 6.31e5·21-s − 5.01e5·25-s + 2.77e5·27-s − 8.71e5·31-s − 2.31e6·37-s − 1.48e6·39-s + 1.98e6·43-s + 1.71e7·49-s + 1.63e7·57-s − 3.87e7·61-s + 2.37e7·63-s − 5.60e7·67-s − 5.04e7·73-s + 5.11e7·75-s + 1.26e8·79-s − 5.34e7·81-s + 9.02e7·91-s + 8.89e7·93-s + 3.91e7·97-s − 2.26e8·103-s − 3.48e6·109-s + 2.36e8·111-s + ⋯ |
| L(s) = 1 | − 1.25·3-s + 2.57·7-s + 0.585·9-s + 0.510·13-s − 1.23·19-s − 3.24·21-s − 1.28·25-s + 0.521·27-s − 0.944·31-s − 1.23·37-s − 0.643·39-s + 0.579·43-s + 2.98·49-s + 1.55·57-s − 2.79·61-s + 1.50·63-s − 2.78·67-s − 1.77·73-s + 1.61·75-s + 3.25·79-s − 1.24·81-s + 1.31·91-s + 1.18·93-s + 0.441·97-s − 2.01·103-s − 0.0247·109-s + 1.55·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.8684003379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8684003379\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 34 p T + p^{8} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 100358 p T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 442 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 38857702 p T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7294 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 10482174722 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 80326 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 147132606722 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 253546712162 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 435914 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1159298 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8591852110082 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 990266 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2717384513282 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 23095221504482 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 291106443928802 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 19369154 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 28024294 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 158683081351682 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 25230142 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 63401398 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2244210903661922 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1743003813196802 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19550306 T + p^{8} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.13100612737944975412304869197, −10.94401677546507450886323530604, −10.66827224770834781316350975370, −10.12657852084085885606667850476, −9.154528807839883410974724699881, −8.756745515124281958583804599176, −8.284025225401463377407647724734, −7.59878234507095300550132126770, −7.49668498089429125020928255597, −6.47770522160151388897061498731, −6.04125369654181492163853462739, −5.49512965138004323519695458990, −5.03802951101396085371967615815, −4.45015129361889265312471426218, −4.18756660338404913032296328300, −3.16392055015717875210103855660, −2.02241841222276603025194389228, −1.70735793443893553717919723476, −1.19581429385025993516723707960, −0.24371974521369785222490174787,
0.24371974521369785222490174787, 1.19581429385025993516723707960, 1.70735793443893553717919723476, 2.02241841222276603025194389228, 3.16392055015717875210103855660, 4.18756660338404913032296328300, 4.45015129361889265312471426218, 5.03802951101396085371967615815, 5.49512965138004323519695458990, 6.04125369654181492163853462739, 6.47770522160151388897061498731, 7.49668498089429125020928255597, 7.59878234507095300550132126770, 8.284025225401463377407647724734, 8.756745515124281958583804599176, 9.154528807839883410974724699881, 10.12657852084085885606667850476, 10.66827224770834781316350975370, 10.94401677546507450886323530604, 11.13100612737944975412304869197
