| L(s) = 1 | − 2·5-s − 9-s − 6·11-s − 4·16-s − 6·19-s − 25-s + 2·29-s + 6·41-s + 2·45-s + 2·49-s + 12·55-s + 2·59-s − 8·61-s + 10·71-s − 2·79-s + 8·80-s − 8·81-s − 14·89-s + 12·95-s + 6·99-s + 2·101-s + 2·109-s + 6·121-s + 12·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/3·9-s − 1.80·11-s − 16-s − 1.37·19-s − 1/5·25-s + 0.371·29-s + 0.937·41-s + 0.298·45-s + 2/7·49-s + 1.61·55-s + 0.260·59-s − 1.02·61-s + 1.18·71-s − 0.225·79-s + 0.894·80-s − 8/9·81-s − 1.48·89-s + 1.23·95-s + 0.603·99-s + 0.199·101-s + 0.191·109-s + 6/11·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 166 T^{2} + p^{2} T^{4} \) |
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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
−10.60322834630461261159775913517, −9.998757408102921271947309103737, −9.375961475465363509033528086061, −8.570813615186415621396555315960, −8.386828057818647069002423247636, −7.70015469156727852843098194525, −7.31786667292571961207119437035, −6.55163036893510120770712569533, −5.90778291006245116087130592935, −5.18008412133703400261189070338, −4.52612972491254181283243678017, −3.95385135452975289417316690141, −2.89288342488853633553795450797, −2.25715038039026452227887201723, 0,
2.25715038039026452227887201723, 2.89288342488853633553795450797, 3.95385135452975289417316690141, 4.52612972491254181283243678017, 5.18008412133703400261189070338, 5.90778291006245116087130592935, 6.55163036893510120770712569533, 7.31786667292571961207119437035, 7.70015469156727852843098194525, 8.386828057818647069002423247636, 8.570813615186415621396555315960, 9.375961475465363509033528086061, 9.998757408102921271947309103737, 10.60322834630461261159775913517
