L(s) = 1 | − 2·4-s − 3·5-s − 9-s − 6·11-s − 19-s + 6·20-s + 4·25-s + 9·29-s − 12·31-s + 2·36-s + 12·44-s + 3·45-s + 2·49-s + 18·55-s + 9·59-s − 19·61-s + 8·64-s + 9·71-s + 2·76-s + 9·79-s − 8·81-s − 9·89-s + 3·95-s + 6·99-s − 8·100-s + 3·101-s − 27·109-s + ⋯ |
L(s) = 1 | − 4-s − 1.34·5-s − 1/3·9-s − 1.80·11-s − 0.229·19-s + 1.34·20-s + 4/5·25-s + 1.67·29-s − 2.15·31-s + 1/3·36-s + 1.80·44-s + 0.447·45-s + 2/7·49-s + 2.42·55-s + 1.17·59-s − 2.43·61-s + 64-s + 1.06·71-s + 0.229·76-s + 1.01·79-s − 8/9·81-s − 0.953·89-s + 0.307·95-s + 0.603·99-s − 4/5·100-s + 0.298·101-s − 2.58·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 142 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
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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
−11.20596463141761880232308913941, −10.86663596298499142634886865184, −10.32692888075682554881955096544, −9.600825932349098521069904813407, −8.902747827759237921715267272211, −8.434380274846306746371413761130, −7.907590442336435129393185114445, −7.46730325217072203305692834103, −6.66583993396032814586673339969, −5.57733380179038841837323712732, −5.03077182024000936944442191827, −4.36902019806103263721464598563, −3.60401610781285747484879091023, −2.63792299456794529108614129708, 0,
2.63792299456794529108614129708, 3.60401610781285747484879091023, 4.36902019806103263721464598563, 5.03077182024000936944442191827, 5.57733380179038841837323712732, 6.66583993396032814586673339969, 7.46730325217072203305692834103, 7.907590442336435129393185114445, 8.434380274846306746371413761130, 8.902747827759237921715267272211, 9.600825932349098521069904813407, 10.32692888075682554881955096544, 10.86663596298499142634886865184, 11.20596463141761880232308913941