| L(s) = 1 | − 2·3-s + 2·5-s − 6·7-s − 4·15-s + 4·19-s + 12·21-s − 6·23-s + 3·25-s + 2·27-s − 12·31-s − 12·35-s − 12·37-s − 12·41-s + 10·43-s − 6·47-s + 16·49-s − 8·57-s + 12·59-s + 12·61-s − 10·67-s + 12·69-s − 12·71-s + 8·73-s − 6·75-s − 24·79-s − 81-s − 6·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 2.26·7-s − 1.03·15-s + 0.917·19-s + 2.61·21-s − 1.25·23-s + 3/5·25-s + 0.384·27-s − 2.15·31-s − 2.02·35-s − 1.97·37-s − 1.87·41-s + 1.52·43-s − 0.875·47-s + 16/7·49-s − 1.05·57-s + 1.56·59-s + 1.53·61-s − 1.22·67-s + 1.44·69-s − 1.42·71-s + 0.936·73-s − 0.692·75-s − 2.70·79-s − 1/9·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 132 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.425599654296558230734142623570, −9.369617944511913742317556790155, −8.788555865680087591935935879703, −8.378416145332349679209526175504, −7.75006451927431289142492906686, −7.05888379108100852942441508436, −6.89030060047958616986861920984, −6.57444068311977131761921957332, −6.08829428817590354920690143168, −5.60663955084435907284790122873, −5.38057180921152281224334499025, −5.28995590108713009777018455637, −4.10912843233860059316949401395, −3.84683090402156849194763064797, −3.17331369713556042964900992061, −2.89206975625698716440177114881, −2.05913253562756355853046760022, −1.38644316172211549871292105859, 0, 0,
1.38644316172211549871292105859, 2.05913253562756355853046760022, 2.89206975625698716440177114881, 3.17331369713556042964900992061, 3.84683090402156849194763064797, 4.10912843233860059316949401395, 5.28995590108713009777018455637, 5.38057180921152281224334499025, 5.60663955084435907284790122873, 6.08829428817590354920690143168, 6.57444068311977131761921957332, 6.89030060047958616986861920984, 7.05888379108100852942441508436, 7.75006451927431289142492906686, 8.378416145332349679209526175504, 8.788555865680087591935935879703, 9.369617944511913742317556790155, 9.425599654296558230734142623570
