L(s) = 1 | + 4·3-s − 2·4-s − 2·7-s + 6·9-s − 8·12-s − 4·13-s + 4·19-s − 8·21-s − 4·23-s − 2·25-s − 4·27-s + 4·28-s − 8·31-s − 12·36-s − 8·37-s − 16·39-s − 6·41-s + 10·43-s − 8·47-s − 11·49-s + 8·52-s + 16·57-s + 18·59-s + 4·61-s − 12·63-s + 8·64-s + 14·67-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 4-s − 0.755·7-s + 2·9-s − 2.30·12-s − 1.10·13-s + 0.917·19-s − 1.74·21-s − 0.834·23-s − 2/5·25-s − 0.769·27-s + 0.755·28-s − 1.43·31-s − 2·36-s − 1.31·37-s − 2.56·39-s − 0.937·41-s + 1.52·43-s − 1.16·47-s − 1.57·49-s + 1.10·52-s + 2.11·57-s + 2.34·59-s + 0.512·61-s − 1.51·63-s + 64-s + 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36132121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36132121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 6011 | | \( 1+O(T) \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 151 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 255 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 171 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.037320087542019907088555948111, −7.80227777642978981045443275430, −7.29086405724718732742521075575, −6.89131336795170439683812530891, −6.83144406496366616624902535991, −6.12411223570190490749209900801, −5.59315384846023000001200001534, −5.33599932400916277552699035183, −5.02718628241489753749585940691, −4.51824382186952005110477732235, −3.92453793980422204078593130716, −3.70934060798752732765978003320, −3.46759832927655197654433517784, −3.13490711696784360345618452452, −2.52817507160489180094504075009, −2.20553243954179432429620854291, −1.97859821165285673217133941978, −1.16736821809428435350262410335, 0, 0,
1.16736821809428435350262410335, 1.97859821165285673217133941978, 2.20553243954179432429620854291, 2.52817507160489180094504075009, 3.13490711696784360345618452452, 3.46759832927655197654433517784, 3.70934060798752732765978003320, 3.92453793980422204078593130716, 4.51824382186952005110477732235, 5.02718628241489753749585940691, 5.33599932400916277552699035183, 5.59315384846023000001200001534, 6.12411223570190490749209900801, 6.83144406496366616624902535991, 6.89131336795170439683812530891, 7.29086405724718732742521075575, 7.80227777642978981045443275430, 8.037320087542019907088555948111