L(s) = 1 | − 2-s − 3-s + 6-s − 2·7-s + 8-s − 2·11-s − 5·13-s + 2·14-s − 16-s + 5·17-s + 2·19-s + 2·21-s + 2·22-s + 6·23-s − 24-s + 5·26-s + 27-s + 9·29-s − 8·31-s + 2·33-s − 5·34-s − 11·37-s − 2·38-s + 5·39-s − 5·41-s − 2·42-s + 10·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 0.603·11-s − 1.38·13-s + 0.534·14-s − 1/4·16-s + 1.21·17-s + 0.458·19-s + 0.436·21-s + 0.426·22-s + 1.25·23-s − 0.204·24-s + 0.980·26-s + 0.192·27-s + 1.67·29-s − 1.43·31-s + 0.348·33-s − 0.857·34-s − 1.80·37-s − 0.324·38-s + 0.800·39-s − 0.780·41-s − 0.308·42-s + 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7427624616\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7427624616\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.677788571556883400528632144562, −9.237522268723670292361054805902, −8.477280762586459727543210948882, −8.259420724833814489426189103957, −7.956542057082898298864694613599, −7.23139481370337506647607902365, −7.05826323122888421135468924390, −6.88412999825637603769377373884, −6.35132326628754040069469738147, −5.47528181638312433633260009252, −5.45293530119668154936591671208, −5.15230332391603957313320796344, −4.68933852967818017784298570556, −3.83959236379475566361257914814, −3.63749792718024526195840527056, −2.80236306209461528046208106546, −2.66949632228976515784812760304, −1.85294822261728006529051524677, −0.991259587611552117330735909982, −0.45308523283961987254618226267,
0.45308523283961987254618226267, 0.991259587611552117330735909982, 1.85294822261728006529051524677, 2.66949632228976515784812760304, 2.80236306209461528046208106546, 3.63749792718024526195840527056, 3.83959236379475566361257914814, 4.68933852967818017784298570556, 5.15230332391603957313320796344, 5.45293530119668154936591671208, 5.47528181638312433633260009252, 6.35132326628754040069469738147, 6.88412999825637603769377373884, 7.05826323122888421135468924390, 7.23139481370337506647607902365, 7.956542057082898298864694613599, 8.259420724833814489426189103957, 8.477280762586459727543210948882, 9.237522268723670292361054805902, 9.677788571556883400528632144562