| L(s) = 1 | + 2·3-s + 2·5-s + 6·7-s + 3·9-s + 4·11-s − 2·13-s + 4·15-s + 6·17-s + 6·19-s + 12·21-s + 10·23-s + 3·25-s + 4·27-s + 14·29-s + 8·33-s + 12·35-s − 8·37-s − 4·39-s − 16·41-s − 8·43-s + 6·45-s + 8·47-s + 18·49-s + 12·51-s − 8·53-s + 8·55-s + 12·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 2.26·7-s + 9-s + 1.20·11-s − 0.554·13-s + 1.03·15-s + 1.45·17-s + 1.37·19-s + 2.61·21-s + 2.08·23-s + 3/5·25-s + 0.769·27-s + 2.59·29-s + 1.39·33-s + 2.02·35-s − 1.31·37-s − 0.640·39-s − 2.49·41-s − 1.21·43-s + 0.894·45-s + 1.16·47-s + 18/7·49-s + 1.68·51-s − 1.09·53-s + 1.07·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.89468663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.89468663\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_4$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T - 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.311481282284072180722282219587, −7.949572919983595512216130281996, −7.53057459290113868298200435676, −7.24041837715562060719884066065, −6.91517243378275692103937686473, −6.65745772343635201272952541200, −6.13305548224059966560884753647, −5.52966501607291543349755353005, −5.26453565800024875436271382449, −4.92644355413436660663163410898, −4.58597725520911454640598224130, −4.56581120146381618493833082419, −3.58662327688077772415580842543, −3.32753085256170151129810779146, −3.03826463500805164461530768240, −2.59857374764109113318411532864, −1.88776491128068717166449546614, −1.60264292572606424632057476284, −1.15856166037910939149156998981, −1.05645825436883446934579532177,
1.05645825436883446934579532177, 1.15856166037910939149156998981, 1.60264292572606424632057476284, 1.88776491128068717166449546614, 2.59857374764109113318411532864, 3.03826463500805164461530768240, 3.32753085256170151129810779146, 3.58662327688077772415580842543, 4.56581120146381618493833082419, 4.58597725520911454640598224130, 4.92644355413436660663163410898, 5.26453565800024875436271382449, 5.52966501607291543349755353005, 6.13305548224059966560884753647, 6.65745772343635201272952541200, 6.91517243378275692103937686473, 7.24041837715562060719884066065, 7.53057459290113868298200435676, 7.949572919983595512216130281996, 8.311481282284072180722282219587
