| L(s) = 1 | + 12·11-s + 7·16-s − 152·31-s − 228·41-s − 112·61-s − 168·71-s + 192·101-s − 394·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 84·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 1.09·11-s + 7/16·16-s − 4.90·31-s − 5.56·41-s − 1.83·61-s − 2.36·71-s + 1.90·101-s − 3.25·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.477·176-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1053998706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1053998706\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 2302 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 120817 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 697 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 338782 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 782 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 132578 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 57 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 6484898 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 8816738 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 2892862 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 20810017 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 49190543 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 6082 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 27517583 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 15617 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - 7798462 T^{4} + p^{8} T^{8} \) |
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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
−8.816064839565592963096325085293, −8.710995006050827522679517770718, −8.018394320683349624310175932110, −7.87686297864578694477561461757, −7.74443982720865783610740897097, −7.35171562604295048599959053234, −7.02211502905865862740468964520, −6.80711956296735100158460333586, −6.65053411589934279969974893977, −6.36828837680378856227475322096, −5.80268265136281967277961176080, −5.72655264966527574820639343107, −5.32998019900926768765505391095, −5.01976207111317051179412876354, −4.97818828669437133914202143160, −4.29146790071985406439510430553, −4.03637687945360938964497067194, −3.58728750946998076273035988448, −3.43481396056161488113640760637, −3.24975858714790882141130346495, −2.61123488384729660036919720633, −1.84883495312584071143165209526, −1.57094550524415515468253757216, −1.53185053427783047779234247959, −0.084607343842383066068466514975,
0.084607343842383066068466514975, 1.53185053427783047779234247959, 1.57094550524415515468253757216, 1.84883495312584071143165209526, 2.61123488384729660036919720633, 3.24975858714790882141130346495, 3.43481396056161488113640760637, 3.58728750946998076273035988448, 4.03637687945360938964497067194, 4.29146790071985406439510430553, 4.97818828669437133914202143160, 5.01976207111317051179412876354, 5.32998019900926768765505391095, 5.72655264966527574820639343107, 5.80268265136281967277961176080, 6.36828837680378856227475322096, 6.65053411589934279969974893977, 6.80711956296735100158460333586, 7.02211502905865862740468964520, 7.35171562604295048599959053234, 7.74443982720865783610740897097, 7.87686297864578694477561461757, 8.018394320683349624310175932110, 8.710995006050827522679517770718, 8.816064839565592963096325085293
