| L(s) = 1 | + 14·3-s + 18·7-s + 98·9-s + 252·21-s + 134·23-s + 378·27-s + 612·29-s − 594·43-s + 602·47-s + 162·49-s + 1.76e3·63-s + 1.09e3·67-s + 1.87e3·69-s + 251·81-s − 154·83-s + 8.56e3·87-s − 2.77e3·89-s − 756·101-s + 2.64e3·103-s + 442·107-s − 2.66e3·121-s + 127-s − 8.31e3·129-s + 131-s + 137-s + 139-s + 8.42e3·141-s + ⋯ |
| L(s) = 1 | + 2.69·3-s + 0.971·7-s + 3.62·9-s + 2.61·21-s + 1.21·23-s + 2.69·27-s + 3.91·29-s − 2.10·43-s + 1.86·47-s + 0.472·49-s + 3.52·63-s + 2.00·67-s + 3.27·69-s + 0.344·81-s − 0.203·83-s + 10.5·87-s − 3.30·89-s − 0.744·101-s + 2.53·103-s + 0.399·107-s − 2·121-s + 0.000698·127-s − 5.67·129-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 5.03·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.76067165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.76067165\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 134 T + 8978 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 306 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 74338 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 594 T + 176418 T^{2} + 594 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 602 T + 181202 T^{2} - 602 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 452342 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1098 T + 602802 T^{2} - 1098 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T + 11858 T^{2} + 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1386 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
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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
−9.808609048281878972578923981327, −9.665100365893924827949272038506, −9.010469382426616817428277542851, −8.557372398874795494845711482973, −8.410943102383548587749600420653, −8.344176940099416007143729431144, −7.67237284512240269463608962625, −7.33014321937830420955061285894, −6.70643146102231395000209422300, −6.51447420771866517429125727037, −5.46672794870478567095195885314, −5.07118577408370535203430362419, −4.35067748268439510042753276613, −4.25024550019534403749302682128, −3.26052586723759459267075886264, −3.12543284921504462390505219026, −2.54810363095590080831875985583, −2.15707497683832617851259185625, −1.38649674374557187296766448970, −0.844886913986003852356878683321,
0.844886913986003852356878683321, 1.38649674374557187296766448970, 2.15707497683832617851259185625, 2.54810363095590080831875985583, 3.12543284921504462390505219026, 3.26052586723759459267075886264, 4.25024550019534403749302682128, 4.35067748268439510042753276613, 5.07118577408370535203430362419, 5.46672794870478567095195885314, 6.51447420771866517429125727037, 6.70643146102231395000209422300, 7.33014321937830420955061285894, 7.67237284512240269463608962625, 8.344176940099416007143729431144, 8.410943102383548587749600420653, 8.557372398874795494845711482973, 9.010469382426616817428277542851, 9.665100365893924827949272038506, 9.808609048281878972578923981327
