| L(s) = 1 | − 30·3-s + 8·4-s + 473·9-s − 36·11-s − 240·12-s − 180·17-s − 106·19-s − 250·25-s − 5.19e3·27-s + 1.08e3·33-s + 3.78e3·36-s + 1.56e3·41-s + 580·43-s − 288·44-s + 1.37e3·49-s + 5.40e3·51-s + 3.18e3·57-s − 2.53e3·59-s − 512·64-s + 210·67-s − 1.44e3·68-s + 430·73-s + 7.50e3·75-s − 848·76-s + 4.45e4·81-s + 2.70e3·83-s − 5.73e3·97-s + ⋯ |
| L(s) = 1 | − 5.77·3-s + 4-s + 17.5·9-s − 0.986·11-s − 5.77·12-s − 2.56·17-s − 1.27·19-s − 2·25-s − 36.9·27-s + 5.69·33-s + 17.5·36-s + 5.96·41-s + 2.05·43-s − 0.986·44-s + 4·49-s + 14.8·51-s + 7.38·57-s − 5.60·59-s − 64-s + 0.382·67-s − 2.56·68-s + 0.689·73-s + 11.5·75-s − 1.27·76-s + 61.0·81-s + 3.57·83-s − 5.99·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04883113134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04883113134\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 106 T + 4377 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2}( 1 + 10 T + 73 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 18 T - 1007 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 90 T + 3187 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 522 T + p^{3} T^{2} )^{2}( 1 - 522 T + 203563 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 290 T + 4593 T^{2} - 290 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 846 T + p^{3} T^{2} )^{2}( 1 + 846 T + 510337 T^{2} + 846 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2}( 1 - 70 T - 295863 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 430 T + p^{3} T^{2} )^{2}( 1 + 430 T - 204117 T^{2} + 430 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 1350 T + 1250713 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1026 T + 347707 T^{2} - 1026 p^{3} T^{3} + p^{6} T^{4} )( 1 + 1026 T + 347707 T^{2} + 1026 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 1910 T + p^{3} T^{2} )^{2}( 1 + 1910 T + 2735427 T^{2} + 1910 p^{3} T^{3} + p^{6} T^{4} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.261046100745366965999788014995, −9.183670077729362426290141135111, −8.386327567555373298228195519524, −7.88005847111022262056954850497, −7.48732497840781851329671398241, −7.44070738083707088687383630554, −7.24760988286818634388153299793, −6.78130908415628598451345151087, −6.29522288419487172039917981142, −6.28685316167026305146690106892, −6.13423272920675123399757615175, −6.02883521505841586941429828895, −5.59291027621362977899646809022, −5.42129627825809164495555938306, −5.21950600442080141213556148925, −4.40693229822365045452711434851, −4.34785756305538294755654788215, −4.32903640161705252884229924615, −4.13243473807524950255251611488, −2.58540356318994207179858464471, −2.38059164690742365448247935711, −1.88310311041807846841929844534, −0.990874539328120541721777719443, −0.62651063538362676232083621367, −0.14669499943518989495243465253,
0.14669499943518989495243465253, 0.62651063538362676232083621367, 0.990874539328120541721777719443, 1.88310311041807846841929844534, 2.38059164690742365448247935711, 2.58540356318994207179858464471, 4.13243473807524950255251611488, 4.32903640161705252884229924615, 4.34785756305538294755654788215, 4.40693229822365045452711434851, 5.21950600442080141213556148925, 5.42129627825809164495555938306, 5.59291027621362977899646809022, 6.02883521505841586941429828895, 6.13423272920675123399757615175, 6.28685316167026305146690106892, 6.29522288419487172039917981142, 6.78130908415628598451345151087, 7.24760988286818634388153299793, 7.44070738083707088687383630554, 7.48732497840781851329671398241, 7.88005847111022262056954850497, 8.386327567555373298228195519524, 9.183670077729362426290141135111, 9.261046100745366965999788014995
