| L(s) = 1 | + 2·11-s − 16-s − 4·89-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 2·176-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 + ⋯ |
| L(s) = 1 | + 2·11-s − 16-s − 4·89-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 2·176-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.402802975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402802975\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.148943117436535000733471307183, −8.897213897461852314347870728101, −8.793860599828825712313843901958, −8.139626582578233471673799587764, −7.87279740722943014216247263140, −7.21442510577589526809801131629, −7.01917485879440151615211308252, −6.50316908836612358804794293084, −6.47194816248568494855302905191, −5.82854231331736859325744871499, −5.43950039264207885793310037888, −4.98190229141573149935883669302, −4.31087894259796945565522230331, −4.14989250529904364092095819238, −3.86423390662753462415709507789, −2.98543465640059473653356308600, −2.89133282975621820332282598464, −1.85598143194806943782795038501, −1.70428658917981024818608436446, −0.830067582061538016029864266451,
0.830067582061538016029864266451, 1.70428658917981024818608436446, 1.85598143194806943782795038501, 2.89133282975621820332282598464, 2.98543465640059473653356308600, 3.86423390662753462415709507789, 4.14989250529904364092095819238, 4.31087894259796945565522230331, 4.98190229141573149935883669302, 5.43950039264207885793310037888, 5.82854231331736859325744871499, 6.47194816248568494855302905191, 6.50316908836612358804794293084, 7.01917485879440151615211308252, 7.21442510577589526809801131629, 7.87279740722943014216247263140, 8.139626582578233471673799587764, 8.793860599828825712313843901958, 8.897213897461852314347870728101, 9.148943117436535000733471307183
