| L(s) = 1 | + 3-s + 3·7-s − 13-s + 3·21-s − 2·25-s − 27-s − 39-s + 43-s + 5·49-s + 61-s + 3·67-s − 2·75-s − 2·79-s − 81-s − 3·91-s − 3·97-s + 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s + 5·147-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 3-s + 3·7-s − 13-s + 3·21-s − 2·25-s − 27-s − 39-s + 43-s + 5·49-s + 61-s + 3·67-s − 2·75-s − 2·79-s − 81-s − 3·91-s − 3·97-s + 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s + 5·147-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.478549268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.478549268\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
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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
−9.349162135002096006809003036393, −8.570991253250151868383222416105, −8.487645459118216843843176447230, −8.101612447301050987904066154650, −8.036880504377731639970719930146, −7.47121187638193623255133366214, −7.23400973143624764323803907074, −6.92572137838990452273564673539, −5.91777541917120383364022307576, −5.80345809106591536221378854028, −5.26485439993429910243432686224, −4.99200654388964607915116731433, −4.50389817841139977668935656357, −4.07142739994042895222895715722, −3.84903011757822454714611796292, −3.04798926946014216688482962529, −2.45163591278318856529247039578, −1.96098300464191485909950230985, −1.93363107643259352758655575256, −1.05283801673542125128765205737,
1.05283801673542125128765205737, 1.93363107643259352758655575256, 1.96098300464191485909950230985, 2.45163591278318856529247039578, 3.04798926946014216688482962529, 3.84903011757822454714611796292, 4.07142739994042895222895715722, 4.50389817841139977668935656357, 4.99200654388964607915116731433, 5.26485439993429910243432686224, 5.80345809106591536221378854028, 5.91777541917120383364022307576, 6.92572137838990452273564673539, 7.23400973143624764323803907074, 7.47121187638193623255133366214, 8.036880504377731639970719930146, 8.101612447301050987904066154650, 8.487645459118216843843176447230, 8.570991253250151868383222416105, 9.349162135002096006809003036393
