| L(s) = 1 | − 2·3-s + 4·5-s + 2·7-s + 3·9-s − 6·11-s − 8·15-s − 8·17-s − 6·19-s − 4·21-s + 2·23-s + 5·25-s − 4·27-s − 2·29-s + 4·31-s + 12·33-s + 8·35-s + 14·37-s + 2·41-s + 2·43-s + 12·45-s + 6·47-s − 8·49-s + 16·51-s − 6·53-s − 24·55-s + 12·57-s − 16·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 0.755·7-s + 9-s − 1.80·11-s − 2.06·15-s − 1.94·17-s − 1.37·19-s − 0.872·21-s + 0.417·23-s + 25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s + 2.08·33-s + 1.35·35-s + 2.30·37-s + 0.312·41-s + 0.304·43-s + 1.78·45-s + 0.875·47-s − 8/7·49-s + 2.24·51-s − 0.824·53-s − 3.23·55-s + 1.58·57-s − 2.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 47 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 3 p T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 207 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 164 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.54809138161434619546420290971, −7.42236257052305934378912313068, −6.65737059939517128006911156560, −6.52766643304503246522182889177, −6.11636260325591896090525449280, −6.07236155478885992200702038540, −5.49672643110538817039225471177, −5.39828670039819067948970479008, −4.73751766104934244951651133522, −4.71221874543091718902656302996, −4.26787147449301197560380364074, −4.07877547754605999136455124686, −2.88932420801284205849653402185, −2.82109530102471290168517646616, −2.39577903564991379108570500924, −1.90859250755270574760087188778, −1.63232394227075698580757337514, −1.08224449129108285091996838074, 0, 0,
1.08224449129108285091996838074, 1.63232394227075698580757337514, 1.90859250755270574760087188778, 2.39577903564991379108570500924, 2.82109530102471290168517646616, 2.88932420801284205849653402185, 4.07877547754605999136455124686, 4.26787147449301197560380364074, 4.71221874543091718902656302996, 4.73751766104934244951651133522, 5.39828670039819067948970479008, 5.49672643110538817039225471177, 6.07236155478885992200702038540, 6.11636260325591896090525449280, 6.52766643304503246522182889177, 6.65737059939517128006911156560, 7.42236257052305934378912313068, 7.54809138161434619546420290971
