| L(s) = 1 | + 3·4-s − 2·5-s + 11-s + 5·16-s − 6·20-s − 25-s + 16·31-s + 3·44-s − 2·49-s − 2·55-s + 8·59-s + 3·64-s − 10·80-s − 12·89-s − 3·100-s + 121-s + 48·124-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s + 0.301·11-s + 5/4·16-s − 1.34·20-s − 1/5·25-s + 2.87·31-s + 0.452·44-s − 2/7·49-s − 0.269·55-s + 1.04·59-s + 3/8·64-s − 1.11·80-s − 1.27·89-s − 0.299·100-s + 1/11·121-s + 4.31·124-s + 1.07·125-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 − 2.57·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.950186345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.950186345\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.60290232847304408135830133156, −7.06357245241558270656855619159, −6.86156063367989210039158772723, −6.46286624390891316695752121741, −6.02045916898048232106941994126, −5.70161555368798107683496311288, −4.97199342804118720933760459970, −4.58491876123190085009592262009, −4.04083673948420490884258308720, −3.64920117372470453410488274693, −2.90668071449851211497557084111, −2.76291904484387861999019027273, −2.06493997072077583121716954000, −1.41914140794236273960317365267, −0.67051293034997416873087953327,
0.67051293034997416873087953327, 1.41914140794236273960317365267, 2.06493997072077583121716954000, 2.76291904484387861999019027273, 2.90668071449851211497557084111, 3.64920117372470453410488274693, 4.04083673948420490884258308720, 4.58491876123190085009592262009, 4.97199342804118720933760459970, 5.70161555368798107683496311288, 6.02045916898048232106941994126, 6.46286624390891316695752121741, 6.86156063367989210039158772723, 7.06357245241558270656855619159, 7.60290232847304408135830133156
