| L(s) = 1 | + 2·5-s − 6·11-s + 2·13-s + 10·19-s − 6·23-s + 3·25-s + 4·29-s + 6·31-s + 8·37-s + 16·41-s + 2·43-s − 16·47-s − 2·49-s − 16·53-s − 12·55-s − 14·59-s − 4·61-s + 4·65-s − 4·67-s + 6·71-s − 12·79-s + 16·83-s − 4·89-s + 20·95-s + 4·97-s + 20·101-s + 10·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.80·11-s + 0.554·13-s + 2.29·19-s − 1.25·23-s + 3/5·25-s + 0.742·29-s + 1.07·31-s + 1.31·37-s + 2.49·41-s + 0.304·43-s − 2.33·47-s − 2/7·49-s − 2.19·53-s − 1.61·55-s − 1.82·59-s − 0.512·61-s + 0.496·65-s − 0.488·67-s + 0.712·71-s − 1.35·79-s + 1.75·83-s − 0.423·89-s + 2.05·95-s + 0.406·97-s + 1.99·101-s + 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.453581228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.453581228\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 84 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 140 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.87136484767984821640057594305, −7.86158064968091233837614148640, −7.18980666214857572846338840966, −6.89069816028572138384546700980, −6.23514270443076353691202059104, −6.21393544001314288317710468639, −5.87808070614564736715970419961, −5.55642396752955825751805436480, −5.02490115997392331637454644386, −4.94772262988711631525525117786, −4.37762044723104436202165789583, −4.26597262333943599412209494631, −3.33457611905444078146084296748, −3.19438627226314528760656744673, −2.76561124681859149091724272804, −2.62400378109031144409736863390, −1.85696068146756143791025419956, −1.58729510428234690086497278736, −0.953520338706024307302501199620, −0.47088910645278908199223118970,
0.47088910645278908199223118970, 0.953520338706024307302501199620, 1.58729510428234690086497278736, 1.85696068146756143791025419956, 2.62400378109031144409736863390, 2.76561124681859149091724272804, 3.19438627226314528760656744673, 3.33457611905444078146084296748, 4.26597262333943599412209494631, 4.37762044723104436202165789583, 4.94772262988711631525525117786, 5.02490115997392331637454644386, 5.55642396752955825751805436480, 5.87808070614564736715970419961, 6.21393544001314288317710468639, 6.23514270443076353691202059104, 6.89069816028572138384546700980, 7.18980666214857572846338840966, 7.86158064968091233837614148640, 7.87136484767984821640057594305
