| L(s) = 1 | + 14·25-s + 28·37-s − 12·53-s − 28·109-s − 48·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 14/5·25-s + 4.60·37-s − 1.64·53-s − 2.68·109-s − 4.51·113-s + 2/11·121-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 + 2.15·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.236714414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.236714414\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 137 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 175 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.41070402845889236620506200437, −5.40410275202455577239138592310, −5.37328300468896328145378727562, −5.14210947896741234214840893680, −4.63610455247356215551677235085, −4.54701490162146204077787482461, −4.46561033357428305403972335795, −4.43903261586478460751474890210, −4.06294150397276758023733919997, −3.96819414719334244870961484934, −3.75023798995059636919243208253, −3.26173176890747296441138653102, −3.15349020679841244817753895901, −3.10821901762098036051915793926, −2.93525286129226448424257684494, −2.50241765438592338457898545696, −2.42002007039666215648604019798, −2.22266205350718119884847835774, −2.21463308052004994395259050842, −1.33651556377193452369080395875, −1.32368125654640021219950217311, −1.21833917302875431270545295753, −1.10769675307881385490225290580, −0.49287259945042488349503557763, −0.29296936738202969112214523879,
0.29296936738202969112214523879, 0.49287259945042488349503557763, 1.10769675307881385490225290580, 1.21833917302875431270545295753, 1.32368125654640021219950217311, 1.33651556377193452369080395875, 2.21463308052004994395259050842, 2.22266205350718119884847835774, 2.42002007039666215648604019798, 2.50241765438592338457898545696, 2.93525286129226448424257684494, 3.10821901762098036051915793926, 3.15349020679841244817753895901, 3.26173176890747296441138653102, 3.75023798995059636919243208253, 3.96819414719334244870961484934, 4.06294150397276758023733919997, 4.43903261586478460751474890210, 4.46561033357428305403972335795, 4.54701490162146204077787482461, 4.63610455247356215551677235085, 5.14210947896741234214840893680, 5.37328300468896328145378727562, 5.40410275202455577239138592310, 5.41070402845889236620506200437
