| L(s) = 1 | − 3-s + 4-s + 2·5-s − 12-s + 3·13-s − 2·15-s − 17-s + 2·20-s + 3·25-s + 27-s − 3·39-s − 47-s + 51-s + 3·52-s − 2·60-s − 64-s + 6·65-s − 68-s − 3·73-s − 3·75-s − 79-s − 81-s − 83-s − 2·85-s − 3·97-s + 3·100-s + 108-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 2·5-s − 12-s + 3·13-s − 2·15-s − 17-s + 2·20-s + 3·25-s + 27-s − 3·39-s − 47-s + 51-s + 3·52-s − 2·60-s − 64-s + 6·65-s − 68-s − 3·73-s − 3·75-s − 79-s − 81-s − 83-s − 2·85-s − 3·97-s + 3·100-s + 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.907371093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.907371093\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.187286126930038455213650910060, −9.179291040763371791126752603730, −8.775422706104966744135415091549, −8.246689150730527971556058386911, −8.161810151030676458203986576107, −7.08843877231934599915486727115, −6.82891119589023998683851487640, −6.70782461752361003388159657240, −6.17050651741043198359451326295, −5.89895402691265616349141089682, −5.75066415523741377470466000794, −5.41827720841477317523960941954, −4.56619330363123953541590790927, −4.44865694576175685041412409581, −3.63234749344303478632032259890, −2.94312829310744692393409048619, −2.78930609763453276351320689692, −1.95545074615478379152446187518, −1.53415641740609884891509966318, −1.17557768724919395806095352179,
1.17557768724919395806095352179, 1.53415641740609884891509966318, 1.95545074615478379152446187518, 2.78930609763453276351320689692, 2.94312829310744692393409048619, 3.63234749344303478632032259890, 4.44865694576175685041412409581, 4.56619330363123953541590790927, 5.41827720841477317523960941954, 5.75066415523741377470466000794, 5.89895402691265616349141089682, 6.17050651741043198359451326295, 6.70782461752361003388159657240, 6.82891119589023998683851487640, 7.08843877231934599915486727115, 8.161810151030676458203986576107, 8.246689150730527971556058386911, 8.775422706104966744135415091549, 9.179291040763371791126752603730, 9.187286126930038455213650910060
