| L(s) = 1 | + 4·43-s + 10·49-s + 44·73-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 0.609·43-s + 10/7·49-s + 5.14·73-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 + 4·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.635572314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.635572314\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 31 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 233 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 353 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 1207 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−6.14485985747862586596925871571, −6.07621264253540802194809574062, −5.95337220230444632964761875156, −5.56090063953097062876307504031, −5.37841888032954181211495018554, −5.28274439860845902907751134672, −5.09789217788922436378541192282, −4.79217652687978378483764597220, −4.71342193320230157002086677878, −4.18998983051658203156450581191, −4.15538560167687370138764961539, −4.07197661409489979236803001191, −3.66346243005887986051454324403, −3.55342644066711340530940741607, −3.30621121854168864509810343879, −2.96584264605352534332875052487, −2.69808551308676508423730719014, −2.55863393629116346040315352645, −2.28742458077828090506225728344, −1.92948198481441846682218511926, −1.69256223010572704638934595979, −1.54101947502222285047409440643, −0.854188338264071426979192051563, −0.61954714604075562841323873197, −0.54364179763579567930289646449,
0.54364179763579567930289646449, 0.61954714604075562841323873197, 0.854188338264071426979192051563, 1.54101947502222285047409440643, 1.69256223010572704638934595979, 1.92948198481441846682218511926, 2.28742458077828090506225728344, 2.55863393629116346040315352645, 2.69808551308676508423730719014, 2.96584264605352534332875052487, 3.30621121854168864509810343879, 3.55342644066711340530940741607, 3.66346243005887986051454324403, 4.07197661409489979236803001191, 4.15538560167687370138764961539, 4.18998983051658203156450581191, 4.71342193320230157002086677878, 4.79217652687978378483764597220, 5.09789217788922436378541192282, 5.28274439860845902907751134672, 5.37841888032954181211495018554, 5.56090063953097062876307504031, 5.95337220230444632964761875156, 6.07621264253540802194809574062, 6.14485985747862586596925871571
