| L(s) = 1 | + 2·4-s + 48·7-s + 2·16-s − 128·25-s + 96·28-s − 80·31-s + 1.02e3·49-s + 32·64-s + 248·73-s + 192·79-s − 72·97-s − 256·100-s − 112·103-s + 96·112-s − 912·121-s − 160·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.16e3·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 48/7·7-s + 1/8·16-s − 5.11·25-s + 24/7·28-s − 2.58·31-s + 20.9·49-s + 1/2·64-s + 3.39·73-s + 2.43·79-s − 0.742·97-s − 2.55·100-s − 1.08·103-s + 6/7·112-s − 7.53·121-s − 1.29·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.88·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.34875196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.34875196\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T^{2} + p T^{4} - p^{5} T^{6} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 64 T^{2} + 86 p^{2} T^{4} + 64 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 12 T + 103 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 456 T^{2} + 81142 T^{4} + 456 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 582 T^{2} + 139819 T^{4} - 582 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 392 T^{2} + 205334 T^{4} - 392 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 858 T^{2} + 360859 T^{4} - 858 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1184 T^{2} + 701702 T^{4} - 1184 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 252 T^{2} + 1041574 T^{4} - 252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 20 T + 906 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2046 T^{2} + 3603955 T^{4} - 2046 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 1228 T^{2} + 5455142 T^{4} + 1228 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 372 T^{2} - 2849402 T^{4} + 372 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 1024 T^{2} + 369222 T^{4} - 1024 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 5748 T^{2} + 22991302 T^{4} + 5748 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 2728 T^{2} + 26089142 T^{4} + 2728 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14038 T^{2} + 76797339 T^{4} - 14038 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 6850 T^{2} + 32587683 T^{4} - 6850 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 3892 T^{2} + 46735782 T^{4} - 3892 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 62 T + 11123 T^{2} - 62 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 48 T + 12283 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 20356 T^{2} + 193548326 T^{4} + 20356 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 12392 T^{2} + 103634582 T^{4} - 12392 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 18 T + 12823 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.36751250372404553961399007835, −5.19574301582067583321217672119, −4.99691269327750318726327741867, −4.93622917920147717247887892072, −4.81528856590780194565594642068, −4.49404282702699567176019643054, −4.48148501121599917487003800440, −4.18092063514891405185947082620, −4.11910466825977897948629031730, −3.90294057934474220364788006673, −3.76614386662994033572047628249, −3.66358026538266707944747699787, −3.55660315928120250603567590713, −3.13421307425443634619829194095, −2.63443137353908418771111834936, −2.53678691883668876322634536472, −2.35169300117927240355623629578, −2.01738807565393690129839082918, −1.92842577630919410244667154409, −1.79444983368095650296312115114, −1.62515203115120789496786062332, −1.47084022450750386882702714353, −1.43711336500855132675023239743, −0.824318849040110571417410589932, −0.30496554667058034169939045768,
0.30496554667058034169939045768, 0.824318849040110571417410589932, 1.43711336500855132675023239743, 1.47084022450750386882702714353, 1.62515203115120789496786062332, 1.79444983368095650296312115114, 1.92842577630919410244667154409, 2.01738807565393690129839082918, 2.35169300117927240355623629578, 2.53678691883668876322634536472, 2.63443137353908418771111834936, 3.13421307425443634619829194095, 3.55660315928120250603567590713, 3.66358026538266707944747699787, 3.76614386662994033572047628249, 3.90294057934474220364788006673, 4.11910466825977897948629031730, 4.18092063514891405185947082620, 4.48148501121599917487003800440, 4.49404282702699567176019643054, 4.81528856590780194565594642068, 4.93622917920147717247887892072, 4.99691269327750318726327741867, 5.19574301582067583321217672119, 5.36751250372404553961399007835
Plot not available for L-functions of degree greater than 10.
