| L(s) = 1 | − 2·4-s + 8·11-s + 3·16-s + 8·19-s + 8·29-s − 16·41-s − 16·44-s + 40·59-s + 8·61-s − 4·64-s + 16·71-s − 16·76-s + 16·79-s + 14·81-s − 8·101-s + 24·109-s − 16·116-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 32·164-s + ⋯ |
| L(s) = 1 | − 4-s + 2.41·11-s + 3/4·16-s + 1.83·19-s + 1.48·29-s − 2.49·41-s − 2.41·44-s + 5.20·59-s + 1.02·61-s − 1/2·64-s + 1.89·71-s − 1.83·76-s + 1.80·79-s + 14/9·81-s − 0.796·101-s + 2.29·109-s − 1.48·116-s + 1.09·121-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 + 2.49·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.862183820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.862183820\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 706 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1030 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2854 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 5366 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 20 T + 216 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14326 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 16546 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 288 T^{2} + 34226 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 9474 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.26652502480383816346371591988, −6.16878573231872667256803188399, −6.12140095851912650593277190011, −5.47740471991652956578594464177, −5.43867255982711025489240370993, −5.32774007321507042114450348124, −5.16194806951121567740626823273, −4.85903324126666250537723549933, −4.81380078792670869640652719493, −4.35679095534201543586701553313, −4.14717672050199002335861964486, −3.97137077215654833861064425118, −3.83505467024691348348920852431, −3.61370844262757165358118933171, −3.34574644365841410024040179447, −3.18008836411010787699930619470, −3.00101854427892507474109576660, −2.58272739885260420126331420069, −2.00489898009576895277026705974, −1.98928027887369660942751943713, −1.92866865948844188178278373139, −1.02931522940193074541380487008, −1.00935103571065460789243477596, −0.947696966719384440904817519222, −0.47487567633079684544706117702,
0.47487567633079684544706117702, 0.947696966719384440904817519222, 1.00935103571065460789243477596, 1.02931522940193074541380487008, 1.92866865948844188178278373139, 1.98928027887369660942751943713, 2.00489898009576895277026705974, 2.58272739885260420126331420069, 3.00101854427892507474109576660, 3.18008836411010787699930619470, 3.34574644365841410024040179447, 3.61370844262757165358118933171, 3.83505467024691348348920852431, 3.97137077215654833861064425118, 4.14717672050199002335861964486, 4.35679095534201543586701553313, 4.81380078792670869640652719493, 4.85903324126666250537723549933, 5.16194806951121567740626823273, 5.32774007321507042114450348124, 5.43867255982711025489240370993, 5.47740471991652956578594464177, 6.12140095851912650593277190011, 6.16878573231872667256803188399, 6.26652502480383816346371591988
