| L(s) = 1 | − 3·9-s − 6·29-s − 6·41-s + 64-s + 6·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-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 | − 3·9-s − 6·29-s − 6·41-s + 64-s + 6·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2725015872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2725015872\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 29 | \( ( 1 + T )^{6} \) |
| good | 2 | \( 1 - T^{6} + T^{12} \) |
| 7 | \( 1 - T^{6} + T^{12} \) |
| 11 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 13 | \( 1 - T^{6} + T^{12} \) |
| 17 | \( 1 - T^{6} + T^{12} \) |
| 19 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 23 | \( ( 1 + T^{2} )^{6} \) |
| 31 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 37 | \( ( 1 + T^{2} )^{6} \) |
| 41 | \( ( 1 + T + T^{2} )^{6} \) |
| 43 | \( ( 1 + T^{2} )^{6} \) |
| 47 | \( 1 - T^{6} + T^{12} \) |
| 53 | \( ( 1 + T^{2} )^{6} \) |
| 59 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 61 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 67 | \( 1 - T^{6} + T^{12} \) |
| 71 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 73 | \( ( 1 + T^{2} )^{6} \) |
| 79 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 83 | \( ( 1 + T^{2} )^{6} \) |
| 89 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 97 | \( ( 1 + T^{2} )^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.02567961140049885902652765560, −4.93310556136923982843168318248, −4.87521113036297526372731130831, −4.54052864524554951160963920347, −4.18238621047646188466607106595, −3.99792092604343512027324767418, −3.92227413147199346103681984728, −3.92038011284746429395361893673, −3.68358561489941601004838613502, −3.43410170924670516091633517555, −3.41521212120751373256652689672, −3.37782493602904219011034578546, −3.08517586013256609681273757950, −2.89900020872895979362590543461, −2.74694215368942342595023696109, −2.63092421854304962046399015711, −2.38606741449672637671138149564, −2.06142696728767120350601762134, −1.88957024790543485481415868619, −1.79161249331612592450483384027, −1.78495838208498490317553901161, −1.56395519216789662685669275993, −1.17871366093912010891378725131, −0.52974695487465657582815957331, −0.30197695048536623300829006736,
0.30197695048536623300829006736, 0.52974695487465657582815957331, 1.17871366093912010891378725131, 1.56395519216789662685669275993, 1.78495838208498490317553901161, 1.79161249331612592450483384027, 1.88957024790543485481415868619, 2.06142696728767120350601762134, 2.38606741449672637671138149564, 2.63092421854304962046399015711, 2.74694215368942342595023696109, 2.89900020872895979362590543461, 3.08517586013256609681273757950, 3.37782493602904219011034578546, 3.41521212120751373256652689672, 3.43410170924670516091633517555, 3.68358561489941601004838613502, 3.92038011284746429395361893673, 3.92227413147199346103681984728, 3.99792092604343512027324767418, 4.18238621047646188466607106595, 4.54052864524554951160963920347, 4.87521113036297526372731130831, 4.93310556136923982843168318248, 5.02567961140049885902652765560
Plot not available for L-functions of degree greater than 10.
