| L(s) = 1 | − 4·5-s + 10·13-s − 10·17-s + 11·25-s − 10·37-s + 16·41-s − 10·53-s + 24·61-s − 40·65-s + 10·73-s − 9·81-s + 40·85-s + 10·97-s − 4·101-s + 30·113-s + 22·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2.77·13-s − 2.42·17-s + 11/5·25-s − 1.64·37-s + 2.49·41-s − 1.37·53-s + 3.07·61-s − 4.96·65-s + 1.17·73-s − 81-s + 4.33·85-s + 1.01·97-s − 0.398·101-s + 2.82·113-s + 2·121-s − 2.14·125-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028080073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028080073\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−11.45823637494045206387330414327, −11.34802083647064143476429057640, −11.17739061108679690381095781286, −10.75088767626022843866068533848, −10.21521912329344702226891179652, −9.290739722645913941443564781581, −8.753983348100711241549479155291, −8.674534544629272867030782860446, −8.203325745555624988059499181647, −7.65282297564612043595752202219, −7.00036321406455903231163002005, −6.59336922864393581770584593663, −6.17080292838263670808391925213, −5.40503843578542021326014796610, −4.55279357045582279894781279974, −4.12396539936214544141183389183, −3.73628229388653838275156087841, −3.12558842572288379734781975579, −2.02547983030528401194618204137, −0.76085770125354409810161120625,
0.76085770125354409810161120625, 2.02547983030528401194618204137, 3.12558842572288379734781975579, 3.73628229388653838275156087841, 4.12396539936214544141183389183, 4.55279357045582279894781279974, 5.40503843578542021326014796610, 6.17080292838263670808391925213, 6.59336922864393581770584593663, 7.00036321406455903231163002005, 7.65282297564612043595752202219, 8.203325745555624988059499181647, 8.674534544629272867030782860446, 8.753983348100711241549479155291, 9.290739722645913941443564781581, 10.21521912329344702226891179652, 10.75088767626022843866068533848, 11.17739061108679690381095781286, 11.34802083647064143476429057640, 11.45823637494045206387330414327
