| L(s) = 1 | − 4·7-s − 16·23-s + 8·25-s + 8·31-s + 16·41-s + 24·47-s + 10·49-s − 16·71-s − 24·73-s + 16·79-s + 2·81-s + 24·89-s + 32·97-s + 40·103-s − 32·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 24·169-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 3.33·23-s + 8/5·25-s + 1.43·31-s + 2.49·41-s + 3.50·47-s + 10/7·49-s − 1.89·71-s − 2.80·73-s + 1.80·79-s + 2/9·81-s + 2.54·89-s + 3.24·97-s + 3.94·103-s − 3.01·113-s − 0.363·121-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 + 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436509823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436509823\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 1726 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4$ | \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7222 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 10078 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 7118 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 25342 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.59021682681677904205425769295, −6.26391838175745026487182060941, −6.20444888506338909861209002397, −5.95281327693716429520513640035, −5.91798504323521121380142921577, −5.67992878438761921294310125418, −5.46120784367117280552094956156, −5.09586236456125854372749258272, −4.81681060798973566215822388612, −4.42802451660718185468943516494, −4.38383469190421564463829189040, −4.36586457338485032119634414987, −3.93357415583211214208073769730, −3.73144865045609170439656549548, −3.37537502221229964077450199679, −3.27434029358543528670025521031, −3.09007272737051820480564957414, −2.51481100218841815462307982856, −2.28299938708097598802377792935, −2.28038867430418944832058444007, −2.22156709279074831659160070744, −1.28263671569330818281342164941, −1.13807936157466568178020860239, −0.75985471698943355254151546823, −0.26549596608519342131853183241,
0.26549596608519342131853183241, 0.75985471698943355254151546823, 1.13807936157466568178020860239, 1.28263671569330818281342164941, 2.22156709279074831659160070744, 2.28038867430418944832058444007, 2.28299938708097598802377792935, 2.51481100218841815462307982856, 3.09007272737051820480564957414, 3.27434029358543528670025521031, 3.37537502221229964077450199679, 3.73144865045609170439656549548, 3.93357415583211214208073769730, 4.36586457338485032119634414987, 4.38383469190421564463829189040, 4.42802451660718185468943516494, 4.81681060798973566215822388612, 5.09586236456125854372749258272, 5.46120784367117280552094956156, 5.67992878438761921294310125418, 5.91798504323521121380142921577, 5.95281327693716429520513640035, 6.20444888506338909861209002397, 6.26391838175745026487182060941, 6.59021682681677904205425769295
