L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 9-s + 4·15-s − 2·19-s − 4·21-s + 25-s + 2·27-s + 2·29-s − 4·35-s − 2·41-s − 2·45-s + 49-s + 2·53-s + 4·57-s − 2·59-s + 2·63-s − 2·75-s + 2·79-s − 4·81-s − 4·87-s + 4·95-s + 8·105-s + 2·107-s + 4·123-s + 2·125-s + ⋯ |
L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 9-s + 4·15-s − 2·19-s − 4·21-s + 25-s + 2·27-s + 2·29-s − 4·35-s − 2·41-s − 2·45-s + 49-s + 2·53-s + 4·57-s − 2·59-s + 2·63-s − 2·75-s + 2·79-s − 4·81-s − 4·87-s + 4·95-s + 8·105-s + 2·107-s + 4·123-s + 2·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2612748008\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2612748008\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 59 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + T^{4} \) |
| 67 | $C_2^2$ | \( 1 + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_2^2$ | \( 1 + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.852089591813853928835309259401, −8.772322615712581420711463191236, −8.598662510147989254445713855650, −8.550951089807183843708187606070, −7.979511113899550973424786685493, −7.71350078067324862489581751442, −7.35133015370066041643390713071, −6.62312195612586303083329015319, −6.42805889550096033045118583514, −6.16127650810643382583918606239, −5.40988162806161428764952057013, −5.03424764790364988943263660364, −4.83250821209533130616843293556, −4.45058467517575565253242398420, −4.07729212326524224375286879471, −3.53632811119413363569362669557, −2.80374303056836814814813590677, −2.08448191437344746400479906503, −1.33244986560295771247592084330, −0.46413613202372031524500103071,
0.46413613202372031524500103071, 1.33244986560295771247592084330, 2.08448191437344746400479906503, 2.80374303056836814814813590677, 3.53632811119413363569362669557, 4.07729212326524224375286879471, 4.45058467517575565253242398420, 4.83250821209533130616843293556, 5.03424764790364988943263660364, 5.40988162806161428764952057013, 6.16127650810643382583918606239, 6.42805889550096033045118583514, 6.62312195612586303083329015319, 7.35133015370066041643390713071, 7.71350078067324862489581751442, 7.979511113899550973424786685493, 8.550951089807183843708187606070, 8.598662510147989254445713855650, 8.772322615712581420711463191236, 9.852089591813853928835309259401