| L(s) = 1 | − 3-s − 2·5-s + 5·7-s − 3·11-s − 2·13-s + 2·15-s + 2·17-s − 5·19-s − 5·21-s − 6·23-s − 7·25-s − 2·27-s + 18·29-s − 8·31-s + 3·33-s − 10·35-s − 8·37-s + 2·39-s − 18·43-s + 2·47-s + 10·49-s − 2·51-s − 11·53-s + 6·55-s + 5·57-s − 8·59-s + 19·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.88·7-s − 0.904·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s − 1.14·19-s − 1.09·21-s − 1.25·23-s − 7/5·25-s − 0.384·27-s + 3.34·29-s − 1.43·31-s + 0.522·33-s − 1.69·35-s − 1.31·37-s + 0.320·39-s − 2.74·43-s + 0.291·47-s + 10/7·49-s − 0.280·51-s − 1.51·53-s + 0.809·55-s + 0.662·57-s − 1.04·59-s + 2.43·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 131 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 113 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 207 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 141 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 153 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 145 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 67 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−8.103052231480263499842354308172, −8.054991676977948232263692444714, −7.892985702703196700993845093024, −7.44084809471158234839169015823, −6.73743193133315918976026983512, −6.71030374719761946720104240538, −6.12875766149546178316463851071, −5.65164537783350556334833270245, −5.10281948304524313975921328591, −4.99482850605282152392250394470, −4.79801781549219476153015376459, −4.13671124984514922475686817104, −3.76230793629053182483183411962, −3.47636578903107047760666768714, −2.55900356052614806103740122534, −2.25749410450786989788373293403, −1.72196835284784521258000404542, −1.22304008262650879874074726100, 0, 0,
1.22304008262650879874074726100, 1.72196835284784521258000404542, 2.25749410450786989788373293403, 2.55900356052614806103740122534, 3.47636578903107047760666768714, 3.76230793629053182483183411962, 4.13671124984514922475686817104, 4.79801781549219476153015376459, 4.99482850605282152392250394470, 5.10281948304524313975921328591, 5.65164537783350556334833270245, 6.12875766149546178316463851071, 6.71030374719761946720104240538, 6.73743193133315918976026983512, 7.44084809471158234839169015823, 7.892985702703196700993845093024, 8.054991676977948232263692444714, 8.103052231480263499842354308172
