| L(s) = 1 | + 4·4-s − 6·11-s + 12·16-s + 8·19-s + 12·29-s + 4·31-s − 6·41-s − 24·44-s + 13·49-s + 24·59-s + 10·61-s + 32·64-s + 18·71-s + 32·76-s + 14·79-s − 30·89-s − 24·101-s + 20·109-s + 48·116-s + 5·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.80·11-s + 3·16-s + 1.83·19-s + 2.22·29-s + 0.718·31-s − 0.937·41-s − 3.61·44-s + 13/7·49-s + 3.12·59-s + 1.28·61-s + 4·64-s + 2.13·71-s + 3.67·76-s + 1.57·79-s − 3.17·89-s − 2.38·101-s + 1.91·109-s + 4.45·116-s + 5/11·121-s + 1.43·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.623624564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.623624564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
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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
−8.527239826750668076739352976436, −8.509021140382467387835576858668, −8.183246905551827455290125212617, −7.75530058644229333108313063361, −7.26262959787466682243617802239, −7.21116725326482928890417389201, −6.58629373050292881625933106129, −6.57491572446149152971830754892, −5.84614422203121001525573684528, −5.48896371185527099656179700822, −5.12960737531167009541858070152, −5.06587779057835608137621377311, −4.05073218470238055137998794026, −3.69837507387096144843993663812, −3.00807082063165288407293863536, −2.82586830360702652945555636247, −2.40728830464588078321785398906, −2.08008494538240381092539366279, −1.08309753007378548757387807657, −0.857910913631849472874066882781,
0.857910913631849472874066882781, 1.08309753007378548757387807657, 2.08008494538240381092539366279, 2.40728830464588078321785398906, 2.82586830360702652945555636247, 3.00807082063165288407293863536, 3.69837507387096144843993663812, 4.05073218470238055137998794026, 5.06587779057835608137621377311, 5.12960737531167009541858070152, 5.48896371185527099656179700822, 5.84614422203121001525573684528, 6.57491572446149152971830754892, 6.58629373050292881625933106129, 7.21116725326482928890417389201, 7.26262959787466682243617802239, 7.75530058644229333108313063361, 8.183246905551827455290125212617, 8.509021140382467387835576858668, 8.527239826750668076739352976436
