| L(s) = 1 | − 4-s + 2·13-s − 3·16-s − 4·19-s − 7·25-s − 16·31-s − 14·37-s + 4·43-s − 2·52-s + 14·61-s + 7·64-s − 20·67-s + 14·73-s + 4·76-s + 4·79-s − 4·97-s + 7·100-s − 16·103-s + 22·109-s − 10·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 14·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.554·13-s − 3/4·16-s − 0.917·19-s − 7/5·25-s − 2.87·31-s − 2.30·37-s + 0.609·43-s − 0.277·52-s + 1.79·61-s + 7/8·64-s − 2.44·67-s + 1.63·73-s + 0.458·76-s + 0.450·79-s − 0.406·97-s + 7/10·100-s − 1.57·103-s + 2.10·109-s − 0.909·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.15·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.351864881282740620795146992606, −8.046073946619554464126667924969, −7.41567786564871440930638373025, −7.08361481449999721694188935183, −7.07155350805120759726058089148, −6.34503351226356896461300523079, −6.01899900172669675147831817346, −5.72750951628917383799341912272, −5.12939503069937808715385834084, −5.06610493216526632792525739921, −4.42704368704245395155838017340, −3.92992721624233989312251724769, −3.67277783861965739710948188494, −3.49392562877687318654810253062, −2.62764991490176986388350922920, −2.07487228904507212952730528348, −1.87285744143315833987391475219, −1.17194717343387249432052540940, 0, 0,
1.17194717343387249432052540940, 1.87285744143315833987391475219, 2.07487228904507212952730528348, 2.62764991490176986388350922920, 3.49392562877687318654810253062, 3.67277783861965739710948188494, 3.92992721624233989312251724769, 4.42704368704245395155838017340, 5.06610493216526632792525739921, 5.12939503069937808715385834084, 5.72750951628917383799341912272, 6.01899900172669675147831817346, 6.34503351226356896461300523079, 7.07155350805120759726058089148, 7.08361481449999721694188935183, 7.41567786564871440930638373025, 8.046073946619554464126667924969, 8.351864881282740620795146992606
