| L(s) = 1 | + 2·5-s + 2·9-s + 4·11-s − 4·19-s − 25-s + 4·29-s − 16·31-s + 4·41-s + 4·45-s − 6·49-s + 8·55-s + 20·59-s − 12·61-s + 8·71-s − 5·81-s + 4·89-s − 8·95-s + 8·99-s − 12·101-s + 20·109-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2/3·9-s + 1.20·11-s − 0.917·19-s − 1/5·25-s + 0.742·29-s − 2.87·31-s + 0.624·41-s + 0.596·45-s − 6/7·49-s + 1.07·55-s + 2.60·59-s − 1.53·61-s + 0.949·71-s − 5/9·81-s + 0.423·89-s − 0.820·95-s + 0.804·99-s − 1.19·101-s + 1.91·109-s − 0.545·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494507497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494507497\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
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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
−10.63205868806947687376151584522, −10.02903962845263956296816426871, −9.610325820479752915128161554525, −9.116461851488262454950794669344, −8.710983449597010051120350438492, −7.966346157839275655786998819682, −7.21349856853454183234034686305, −6.80685216687329560784577020357, −6.17347444353101165016767123962, −5.66940960198070682579828308583, −4.90472667772212871688171740074, −4.09321441703235896059660907164, −3.57735463172639924942421110349, −2.28237866716572804484448288641, −1.53239042869583881003940329862,
1.53239042869583881003940329862, 2.28237866716572804484448288641, 3.57735463172639924942421110349, 4.09321441703235896059660907164, 4.90472667772212871688171740074, 5.66940960198070682579828308583, 6.17347444353101165016767123962, 6.80685216687329560784577020357, 7.21349856853454183234034686305, 7.966346157839275655786998819682, 8.710983449597010051120350438492, 9.116461851488262454950794669344, 9.610325820479752915128161554525, 10.02903962845263956296816426871, 10.63205868806947687376151584522
