L(s) = 1 | − 3·4-s − 5-s + 8·11-s + 5·16-s + 8·19-s + 3·20-s + 25-s + 4·29-s − 20·41-s − 24·44-s − 14·49-s − 8·55-s + 8·59-s − 4·61-s − 3·64-s + 16·71-s − 24·76-s − 5·80-s + 12·89-s − 8·95-s − 3·100-s − 12·101-s + 28·109-s − 12·116-s + 26·121-s − 125-s + 127-s + ⋯ |
L(s) = 1 | − 3/2·4-s − 0.447·5-s + 2.41·11-s + 5/4·16-s + 1.83·19-s + 0.670·20-s + 1/5·25-s + 0.742·29-s − 3.12·41-s − 3.61·44-s − 2·49-s − 1.07·55-s + 1.04·59-s − 0.512·61-s − 3/8·64-s + 1.89·71-s − 2.75·76-s − 0.559·80-s + 1.27·89-s − 0.820·95-s − 0.299·100-s − 1.19·101-s + 2.68·109-s − 1.11·116-s + 2.36·121-s − 0.0894·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7596636170\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7596636170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−11.57318950916085697381955699128, −11.19897971912574422599178738873, −10.04517050900761317536685250745, −9.738851148469479131038939478590, −9.327820441539330420804608673051, −8.554588868470699894187777077269, −8.455181921164028207665051683361, −7.46261028773657489832215471574, −6.76955885158912340897756950945, −6.20681526904542088835730053039, −5.04860090093668311608273372623, −4.79492945983632365647604814265, −3.63412818236731287070311550143, −3.55714635244582915245852508954, −1.27025970316177507740293768624,
1.27025970316177507740293768624, 3.55714635244582915245852508954, 3.63412818236731287070311550143, 4.79492945983632365647604814265, 5.04860090093668311608273372623, 6.20681526904542088835730053039, 6.76955885158912340897756950945, 7.46261028773657489832215471574, 8.455181921164028207665051683361, 8.554588868470699894187777077269, 9.327820441539330420804608673051, 9.738851148469479131038939478590, 10.04517050900761317536685250745, 11.19897971912574422599178738873, 11.57318950916085697381955699128