| L(s) = 1 | + 9-s − 2·11-s + 2·17-s + 2·19-s + 2·25-s + 18·41-s − 4·49-s − 10·59-s + 10·67-s + 10·73-s + 81-s + 10·83-s + 16·97-s − 2·99-s + 14·107-s − 8·113-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 0.603·11-s + 0.485·17-s + 0.458·19-s + 2/5·25-s + 2.81·41-s − 4/7·49-s − 1.30·59-s + 1.22·67-s + 1.17·73-s + 1/9·81-s + 1.09·83-s + 1.62·97-s − 0.201·99-s + 1.35·107-s − 0.752·113-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.318026149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.318026149\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.65138970068316520169274208122, −7.41344476120979599561029549839, −6.65987846222059028330589857382, −6.44233370210268704199303136881, −5.92751648806593283104940032469, −5.47221555562003164936695535027, −5.07957104961909259865275916939, −4.63044282258574204202711352742, −4.17334756799351199346721933633, −3.59764932905173646531577980939, −3.17142358177213165330718657673, −2.55576760511438227150196852659, −2.12673894706842741447027858501, −1.27239363801546632208526259334, −0.64212835429229504182949172227,
0.64212835429229504182949172227, 1.27239363801546632208526259334, 2.12673894706842741447027858501, 2.55576760511438227150196852659, 3.17142358177213165330718657673, 3.59764932905173646531577980939, 4.17334756799351199346721933633, 4.63044282258574204202711352742, 5.07957104961909259865275916939, 5.47221555562003164936695535027, 5.92751648806593283104940032469, 6.44233370210268704199303136881, 6.65987846222059028330589857382, 7.41344476120979599561029549839, 7.65138970068316520169274208122
