L(s) = 1 | + 4-s + 2·9-s − 2·19-s − 2·31-s + 2·36-s − 2·41-s + 49-s + 2·59-s − 64-s + 2·71-s − 2·76-s + 3·81-s + 2·101-s − 2·109-s + 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s + 2·169-s + ⋯ |
L(s) = 1 | + 4-s + 2·9-s − 2·19-s − 2·31-s + 2·36-s − 2·41-s + 49-s + 2·59-s − 64-s + 2·71-s − 2·76-s + 3·81-s + 2·101-s − 2·109-s + 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.226635758\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.226635758\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−10.60198222553491738499943501981, −10.48296094574690901419543687142, −9.936768971150411729091678009578, −9.623509709002433401868198565158, −8.935659138547500208168029283980, −8.697667552476901436637523884576, −8.124827019566218910674591906819, −7.40180487878064152861123052266, −7.39319484456533031912666756805, −6.66724582494043593124821260800, −6.63987312362749575996215691703, −6.13037179539428549648806427794, −5.21805494329496496428835159761, −5.04714880886659193247391087163, −4.13136210575692824492621943906, −3.98666071039928900341649711891, −3.39112788915764216126170731891, −2.19562916259559998164863113647, −2.17650365835593031561178200114, −1.39342324416269898907440800239,
1.39342324416269898907440800239, 2.17650365835593031561178200114, 2.19562916259559998164863113647, 3.39112788915764216126170731891, 3.98666071039928900341649711891, 4.13136210575692824492621943906, 5.04714880886659193247391087163, 5.21805494329496496428835159761, 6.13037179539428549648806427794, 6.63987312362749575996215691703, 6.66724582494043593124821260800, 7.39319484456533031912666756805, 7.40180487878064152861123052266, 8.124827019566218910674591906819, 8.697667552476901436637523884576, 8.935659138547500208168029283980, 9.623509709002433401868198565158, 9.936768971150411729091678009578, 10.48296094574690901419543687142, 10.60198222553491738499943501981