| L(s) = 1 | + 2·7-s − 4·13-s − 6·17-s − 2·19-s + 12·29-s − 2·31-s + 2·37-s + 12·41-s − 10·43-s − 6·47-s − 8·49-s − 18·53-s + 12·59-s − 8·61-s + 8·67-s + 12·71-s − 10·73-s − 8·79-s − 6·83-s − 8·91-s + 2·97-s + 12·101-s − 16·103-s − 2·109-s − 18·113-s − 12·119-s − 19·121-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 2.22·29-s − 0.359·31-s + 0.328·37-s + 1.87·41-s − 1.52·43-s − 0.875·47-s − 8/7·49-s − 2.47·53-s + 1.56·59-s − 1.02·61-s + 0.977·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s − 0.658·83-s − 0.838·91-s + 0.203·97-s + 1.19·101-s − 1.57·103-s − 0.191·109-s − 1.69·113-s − 1.10·119-s − 1.72·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 144 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 192 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.52823112670035452477514657938, −7.51284282937636357077512343951, −6.83311705257973475996541678650, −6.61797779700303150396046229557, −6.30170526659815313556525023665, −6.17407753021035478280464558705, −5.34575602940067639386150980585, −5.17677881684802930378837505420, −4.72091041102345550663197415681, −4.61434920652647565414799601207, −4.26832180120372390562998740704, −3.72626857043563210723674310832, −3.29510237553682378587951643323, −2.67166675170214527963382748820, −2.49163737578087806360969116730, −2.14842252761752349386239190740, −1.31218399016290879851262057647, −1.28578020329537963669087552742, 0, 0,
1.28578020329537963669087552742, 1.31218399016290879851262057647, 2.14842252761752349386239190740, 2.49163737578087806360969116730, 2.67166675170214527963382748820, 3.29510237553682378587951643323, 3.72626857043563210723674310832, 4.26832180120372390562998740704, 4.61434920652647565414799601207, 4.72091041102345550663197415681, 5.17677881684802930378837505420, 5.34575602940067639386150980585, 6.17407753021035478280464558705, 6.30170526659815313556525023665, 6.61797779700303150396046229557, 6.83311705257973475996541678650, 7.51284282937636357077512343951, 7.52823112670035452477514657938
