L(s) = 1 | − 24·11-s + 28·25-s + 146·49-s − 72·59-s + 100·73-s − 480·83-s − 340·97-s + 600·107-s − 124·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 622·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | − 2.18·11-s + 1.11·25-s + 2.97·49-s − 1.22·59-s + 1.36·73-s − 5.78·83-s − 3.50·97-s + 5.60·107-s − 1.02·121-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 + 3.68·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.460651598\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.460651598\) |
\(L(2)\) |
|
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2}( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 73 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 311 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 394 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 47 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 386 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1246 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2063 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3446 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2018 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 1367 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2903 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 11521 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 8458 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 85 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.28042587178737463444690446627, −6.21961099049867555328505585935, −6.00053446389298124461394930974, −5.71446948665206574661276680631, −5.54458868126134222094911766525, −5.45057809320397448019477638292, −5.22623545635728580617380448273, −4.85877303075578878125965246808, −4.78972685707122645381240121856, −4.44488614821087089234793691291, −4.33949073705684135655385436778, −3.99925808550952381398336226064, −3.88681997272576016016695975114, −3.38190525024980908477156207525, −3.23455318406256749742760258440, −2.92957461216666183253174228215, −2.70826107771489295045035044382, −2.64981101782922160338562815114, −2.14702583419387544047571454795, −2.13488086233835796885912202220, −1.57975058681682120997917239735, −1.28730473824218973490140394386, −0.960563459503952240440666354666, −0.44802356264953959867370377858, −0.29652002863867686158134101331,
0.29652002863867686158134101331, 0.44802356264953959867370377858, 0.960563459503952240440666354666, 1.28730473824218973490140394386, 1.57975058681682120997917239735, 2.13488086233835796885912202220, 2.14702583419387544047571454795, 2.64981101782922160338562815114, 2.70826107771489295045035044382, 2.92957461216666183253174228215, 3.23455318406256749742760258440, 3.38190525024980908477156207525, 3.88681997272576016016695975114, 3.99925808550952381398336226064, 4.33949073705684135655385436778, 4.44488614821087089234793691291, 4.78972685707122645381240121856, 4.85877303075578878125965246808, 5.22623545635728580617380448273, 5.45057809320397448019477638292, 5.54458868126134222094911766525, 5.71446948665206574661276680631, 6.00053446389298124461394930974, 6.21961099049867555328505585935, 6.28042587178737463444690446627