| L(s) = 1 | + 24·23-s + 4·25-s − 24·47-s + 26·49-s − 48·71-s + 28·73-s + 68·97-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 5.00·23-s + 4/5·25-s − 3.50·47-s + 26/7·49-s − 5.69·71-s + 3.27·73-s + 6.90·97-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.122932627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.122932627\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{4} \) |
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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
−6.68657196977784461719239264492, −6.53514936890008132507461309304, −6.07838830049888285149430548222, −6.00911264856131501552894052451, −5.95507492455417505172803172781, −5.46443191799870315769322765718, −5.21908949776217737606825001635, −5.16996613975377823707311410468, −4.81856016857525237650458722872, −4.75482826083617409821092907505, −4.60268467437393158010238812071, −4.31450641160937545432908950424, −4.05128874523655031582852127976, −3.45841452620319856599214902966, −3.42153496309350859207627016320, −3.23331775870519764590869316462, −3.07324744743161453129335164902, −2.88088642188079944365725878460, −2.36511194928738506578203382083, −2.14304868354569730507144067699, −2.01499268218046696378537303760, −1.23577831232377122115055415230, −1.10658455339569368592407642258, −1.00686766085879796991935640853, −0.40103888639450773329049022387,
0.40103888639450773329049022387, 1.00686766085879796991935640853, 1.10658455339569368592407642258, 1.23577831232377122115055415230, 2.01499268218046696378537303760, 2.14304868354569730507144067699, 2.36511194928738506578203382083, 2.88088642188079944365725878460, 3.07324744743161453129335164902, 3.23331775870519764590869316462, 3.42153496309350859207627016320, 3.45841452620319856599214902966, 4.05128874523655031582852127976, 4.31450641160937545432908950424, 4.60268467437393158010238812071, 4.75482826083617409821092907505, 4.81856016857525237650458722872, 5.16996613975377823707311410468, 5.21908949776217737606825001635, 5.46443191799870315769322765718, 5.95507492455417505172803172781, 6.00911264856131501552894052451, 6.07838830049888285149430548222, 6.53514936890008132507461309304, 6.68657196977784461719239264492
