| L(s) = 1 | − 4·7-s − 4·13-s − 12·19-s + 4·25-s + 8·31-s + 18·37-s + 12·43-s + 2·49-s + 8·61-s + 14·73-s − 8·79-s + 16·91-s − 10·97-s − 20·103-s − 18·109-s − 8·121-s + 127-s + 131-s + 48·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.10·13-s − 2.75·19-s + 4/5·25-s + 1.43·31-s + 2.95·37-s + 1.82·43-s + 2/7·49-s + 1.02·61-s + 1.63·73-s − 0.900·79-s + 1.67·91-s − 1.01·97-s − 1.97·103-s − 1.72·109-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-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 + 3/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.011041589105212816329112188166, −7.58435567064855486686723796098, −6.83783779886561071593467447761, −6.67158334968123005796447784320, −6.25787428435554820454968357998, −5.95287424577545614572814364452, −5.24936749392889093682839921044, −4.63640353885541368604350705184, −4.13376904574689612252138819680, −3.98331010642415475247011050545, −2.86108699411148756398142603227, −2.68604415269591467528290082955, −2.21756293670217156757659842930, −0.933841790213200869561946347150, 0,
0.933841790213200869561946347150, 2.21756293670217156757659842930, 2.68604415269591467528290082955, 2.86108699411148756398142603227, 3.98331010642415475247011050545, 4.13376904574689612252138819680, 4.63640353885541368604350705184, 5.24936749392889093682839921044, 5.95287424577545614572814364452, 6.25787428435554820454968357998, 6.67158334968123005796447784320, 6.83783779886561071593467447761, 7.58435567064855486686723796098, 8.011041589105212816329112188166
