L(s) = 1 | − 6·3-s − 4·4-s − 2·7-s + 27·9-s + 24·12-s − 23·13-s − 26·19-s + 12·21-s + 50·25-s − 108·27-s + 8·28-s + 13·31-s − 108·36-s − 26·37-s + 138·39-s − 122·43-s + 49·49-s + 92·52-s + 156·57-s − 47·61-s − 54·63-s + 64·64-s − 109·67-s − 143·73-s − 300·75-s + 104·76-s − 131·79-s + ⋯ |
L(s) = 1 | − 2·3-s − 4-s − 2/7·7-s + 3·9-s + 2·12-s − 1.76·13-s − 1.36·19-s + 4/7·21-s + 2·25-s − 4·27-s + 2/7·28-s + 0.419·31-s − 3·36-s − 0.702·37-s + 3.53·39-s − 2.83·43-s + 49-s + 1.76·52-s + 2.73·57-s − 0.770·61-s − 6/7·63-s + 64-s − 1.62·67-s − 1.95·73-s − 4·75-s + 1.36·76-s − 1.65·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40401 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1309819644\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1309819644\) |
\(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 67 | $C_2$ | \( 1 + 109 T + p^{2} T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 73 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.37989785180066436827984156223, −12.12590869541105093339618042409, −11.53511166745489115658758968433, −11.02598778836723771974596784903, −10.42346652910354195625259955906, −10.07237000929381844761074315513, −9.875216597608155323357458796976, −8.942703347217719822379619944812, −8.732509847723348679328551233261, −7.77138383248080587331601140001, −7.05300930626831523457829107869, −6.80832184686532910289846919800, −6.26222463153244787473063895177, −5.40100923865472497282781521346, −5.06127290966401478386616742807, −4.50301043024326037944885782471, −4.20568913151062147393486670261, −2.90356805260228794966888239561, −1.59432626674497962336218884986, −0.23989960814668240457223262795,
0.23989960814668240457223262795, 1.59432626674497962336218884986, 2.90356805260228794966888239561, 4.20568913151062147393486670261, 4.50301043024326037944885782471, 5.06127290966401478386616742807, 5.40100923865472497282781521346, 6.26222463153244787473063895177, 6.80832184686532910289846919800, 7.05300930626831523457829107869, 7.77138383248080587331601140001, 8.732509847723348679328551233261, 8.942703347217719822379619944812, 9.875216597608155323357458796976, 10.07237000929381844761074315513, 10.42346652910354195625259955906, 11.02598778836723771974596784903, 11.53511166745489115658758968433, 12.12590869541105093339618042409, 12.37989785180066436827984156223