| L(s) = 1 | − 2·2-s + 3·4-s + 4·7-s − 4·8-s + 4·13-s − 8·14-s + 5·16-s + 10·19-s − 8·26-s + 12·28-s + 4·31-s − 6·32-s − 8·37-s − 20·38-s + 18·41-s + 10·43-s − 12·47-s + 4·49-s + 12·52-s − 12·53-s − 16·56-s − 6·59-s + 16·61-s − 8·62-s + 7·64-s − 14·67-s + 12·71-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.51·7-s − 1.41·8-s + 1.10·13-s − 2.13·14-s + 5/4·16-s + 2.29·19-s − 1.56·26-s + 2.26·28-s + 0.718·31-s − 1.06·32-s − 1.31·37-s − 3.24·38-s + 2.81·41-s + 1.52·43-s − 1.75·47-s + 4/7·49-s + 1.66·52-s − 1.64·53-s − 2.13·56-s − 0.781·59-s + 2.04·61-s − 1.01·62-s + 7/8·64-s − 1.71·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.346904419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346904419\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 105 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 121 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 169 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.565122914265422413180519727831, −8.236331521113562060104445980852, −7.84280256884599765358098708901, −7.79560081188642338553291523229, −7.27353266277905106986250896092, −7.09917935813281311790445618491, −6.34152847393911674712762172820, −6.28586800201560105273862277356, −5.61616455572404207731135870286, −5.43891589921856121528684256221, −4.95828145449799586796628422329, −4.50019367787690069354192185985, −4.01923011670138581226104531694, −3.50140165235192517846193406946, −2.94195459979969887835445949365, −2.72717946068575890670220038511, −1.86452554272646001298394071269, −1.58123396705195023043790620993, −1.08780858008953016579553312217, −0.65850855089594296065068152211,
0.65850855089594296065068152211, 1.08780858008953016579553312217, 1.58123396705195023043790620993, 1.86452554272646001298394071269, 2.72717946068575890670220038511, 2.94195459979969887835445949365, 3.50140165235192517846193406946, 4.01923011670138581226104531694, 4.50019367787690069354192185985, 4.95828145449799586796628422329, 5.43891589921856121528684256221, 5.61616455572404207731135870286, 6.28586800201560105273862277356, 6.34152847393911674712762172820, 7.09917935813281311790445618491, 7.27353266277905106986250896092, 7.79560081188642338553291523229, 7.84280256884599765358098708901, 8.236331521113562060104445980852, 8.565122914265422413180519727831
