| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 2·9-s − 4·11-s − 4·15-s − 4·21-s + 14·23-s + 3·25-s − 6·27-s + 4·29-s − 4·31-s + 8·33-s + 4·35-s + 4·37-s + 16·41-s − 2·43-s + 4·45-s + 10·47-s − 6·49-s + 8·53-s − 8·55-s − 8·59-s + 12·61-s + 4·63-s + 2·67-s − 28·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 2/3·9-s − 1.20·11-s − 1.03·15-s − 0.872·21-s + 2.91·23-s + 3/5·25-s − 1.15·27-s + 0.742·29-s − 0.718·31-s + 1.39·33-s + 0.676·35-s + 0.657·37-s + 2.49·41-s − 0.304·43-s + 0.596·45-s + 1.45·47-s − 6/7·49-s + 1.09·53-s − 1.07·55-s − 1.04·59-s + 1.53·61-s + 0.503·63-s + 0.244·67-s − 3.37·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.559774733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559774733\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 14 T + 90 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_4$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 130 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 318 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−10.85792068308556767628727704180, −10.68948727444084471056636163109, −9.901576621996484299546080074340, −9.569409000430326125414644515015, −9.219553187397612582205693299886, −8.577730581705478117191365758372, −8.258695077471144645282983744202, −7.50103128756009181112300889173, −7.22731976147435613324510991079, −6.86686743749977191692739932639, −5.99110522947809411766132989854, −5.88714820152215257205762109896, −5.17288427919653093463025589456, −5.14350547470648856713234435606, −4.51543908439966970385180235961, −3.86452999384309230528522508444, −2.70573982561075479971090872973, −2.63703929835137922226091207661, −1.51088602535737684898134545081, −0.78775872753174730960372480096,
0.78775872753174730960372480096, 1.51088602535737684898134545081, 2.63703929835137922226091207661, 2.70573982561075479971090872973, 3.86452999384309230528522508444, 4.51543908439966970385180235961, 5.14350547470648856713234435606, 5.17288427919653093463025589456, 5.88714820152215257205762109896, 5.99110522947809411766132989854, 6.86686743749977191692739932639, 7.22731976147435613324510991079, 7.50103128756009181112300889173, 8.258695077471144645282983744202, 8.577730581705478117191365758372, 9.219553187397612582205693299886, 9.569409000430326125414644515015, 9.901576621996484299546080074340, 10.68948727444084471056636163109, 10.85792068308556767628727704180
