L(s) = 1 | + 3·3-s + 4-s − 7-s + 6·9-s + 3·12-s + 16-s + 4·19-s − 3·21-s + 7·25-s + 9·27-s − 28-s + 2·31-s + 6·36-s − 2·37-s + 3·48-s + 49-s + 12·57-s − 13·61-s − 6·63-s + 64-s + 21·67-s − 4·73-s + 21·75-s + 4·76-s − 10·79-s + 9·81-s − 3·84-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1/2·4-s − 0.377·7-s + 2·9-s + 0.866·12-s + 1/4·16-s + 0.917·19-s − 0.654·21-s + 7/5·25-s + 1.73·27-s − 0.188·28-s + 0.359·31-s + 36-s − 0.328·37-s + 0.433·48-s + 1/7·49-s + 1.58·57-s − 1.66·61-s − 0.755·63-s + 1/8·64-s + 2.56·67-s − 0.468·73-s + 2.42·75-s + 0.458·76-s − 1.12·79-s + 81-s − 0.327·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.899444515\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.899444515\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 5 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−8.212940248075370897129165098077, −7.80348401917433760724990858699, −7.29623744391917286086061549651, −7.04559011370954116880459693422, −6.59557016596219740830159217958, −6.07854688706901528173685618462, −5.43212099226806004781363222020, −4.90765128538370930874620909369, −4.36535165530639452638801602593, −3.77487566333540990485729092329, −3.23435846117427372584287979412, −2.95305606980784467193837350316, −2.40973903976661527989271613546, −1.73692949903066074477967300274, −0.998831840589346850229603562541,
0.998831840589346850229603562541, 1.73692949903066074477967300274, 2.40973903976661527989271613546, 2.95305606980784467193837350316, 3.23435846117427372584287979412, 3.77487566333540990485729092329, 4.36535165530639452638801602593, 4.90765128538370930874620909369, 5.43212099226806004781363222020, 6.07854688706901528173685618462, 6.59557016596219740830159217958, 7.04559011370954116880459693422, 7.29623744391917286086061549651, 7.80348401917433760724990858699, 8.212940248075370897129165098077