| L(s) = 1 | − 2·4-s − 5·9-s − 6·11-s + 4·13-s + 4·16-s + 25-s − 6·29-s − 12·31-s + 10·36-s − 8·37-s − 12·41-s − 12·43-s + 12·44-s + 12·47-s + 49-s − 8·52-s + 12·53-s − 12·59-s + 16·61-s − 8·64-s − 12·67-s + 16·73-s − 6·79-s + 16·81-s + 24·83-s + 16·97-s + 30·99-s + ⋯ |
| L(s) = 1 | − 4-s − 5/3·9-s − 1.80·11-s + 1.10·13-s + 16-s + 1/5·25-s − 1.11·29-s − 2.15·31-s + 5/3·36-s − 1.31·37-s − 1.87·41-s − 1.82·43-s + 1.80·44-s + 1.75·47-s + 1/7·49-s − 1.10·52-s + 1.64·53-s − 1.56·59-s + 2.04·61-s − 64-s − 1.46·67-s + 1.87·73-s − 0.675·79-s + 16/9·81-s + 2.63·83-s + 1.62·97-s + 3.01·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−16.0761521293, −15.2725263569, −15.2025804258, −14.4884310232, −14.0962101507, −13.4088539635, −13.3483260723, −12.9185255514, −12.0267741459, −11.8074747577, −10.9005640016, −10.6732199132, −10.2617561341, −9.37517559110, −8.75360307520, −8.67949885325, −7.98094729632, −7.50080687239, −6.60075882434, −5.59805653955, −5.47907564041, −4.97734488096, −3.61689003046, −3.40261805033, −2.16340813030, 0,
2.16340813030, 3.40261805033, 3.61689003046, 4.97734488096, 5.47907564041, 5.59805653955, 6.60075882434, 7.50080687239, 7.98094729632, 8.67949885325, 8.75360307520, 9.37517559110, 10.2617561341, 10.6732199132, 10.9005640016, 11.8074747577, 12.0267741459, 12.9185255514, 13.3483260723, 13.4088539635, 14.0962101507, 14.4884310232, 15.2025804258, 15.2725263569, 16.0761521293
