L(s) = 1 | − 9-s − 4·11-s − 14·19-s − 4·29-s − 10·31-s + 24·41-s + 5·49-s − 12·59-s + 26·61-s − 8·71-s + 16·79-s + 81-s − 32·89-s + 4·99-s + 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 14·171-s + ⋯ |
L(s) = 1 | − 1/3·9-s − 1.20·11-s − 3.21·19-s − 0.742·29-s − 1.79·31-s + 3.74·41-s + 5/7·49-s − 1.56·59-s + 3.32·61-s − 0.949·71-s + 1.80·79-s + 1/9·81-s − 3.39·89-s + 0.402·99-s + 1.72·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s + 1.07·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5473711105\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5473711105\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.382180679225378054065618073833, −8.251711654258807428969878703982, −7.61960859333694318278678758499, −7.48864256819948257202563261604, −7.13719852191543241130875719535, −6.48954271962727643542959951163, −6.39676063706975165436882843172, −5.81477532772943756049781928814, −5.53724449978106674949920861592, −5.39102918851410175269321190964, −4.62142884779934403994223876580, −4.35569055707554968331559678215, −3.99056034914336343200142618524, −3.71503505122769703031839500594, −2.94736955497131800590406989821, −2.57417481469800850481440410396, −2.06766597255283751538373498990, −2.03726676943916773429991317158, −0.987627823977419018255840836011, −0.21998672096031077742442897395,
0.21998672096031077742442897395, 0.987627823977419018255840836011, 2.03726676943916773429991317158, 2.06766597255283751538373498990, 2.57417481469800850481440410396, 2.94736955497131800590406989821, 3.71503505122769703031839500594, 3.99056034914336343200142618524, 4.35569055707554968331559678215, 4.62142884779934403994223876580, 5.39102918851410175269321190964, 5.53724449978106674949920861592, 5.81477532772943756049781928814, 6.39676063706975165436882843172, 6.48954271962727643542959951163, 7.13719852191543241130875719535, 7.48864256819948257202563261604, 7.61960859333694318278678758499, 8.251711654258807428969878703982, 8.382180679225378054065618073833