L(s) = 1 | − 4-s + 2·9-s − 8·11-s + 16-s − 12·19-s + 12·29-s − 8·31-s − 2·36-s − 8·41-s + 8·44-s − 28·59-s + 20·61-s − 64-s + 24·71-s + 12·76-s − 8·79-s − 5·81-s + 16·89-s − 16·99-s + 4·101-s − 20·109-s − 12·116-s + 26·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 2/3·9-s − 2.41·11-s + 1/4·16-s − 2.75·19-s + 2.22·29-s − 1.43·31-s − 1/3·36-s − 1.24·41-s + 1.20·44-s − 3.64·59-s + 2.56·61-s − 1/8·64-s + 2.84·71-s + 1.37·76-s − 0.900·79-s − 5/9·81-s + 1.69·89-s − 1.60·99-s + 0.398·101-s − 1.91·109-s − 1.11·116-s + 2.36·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2641808511\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2641808511\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.014619622579347709460126090766, −8.728233298460355061167773961378, −8.178980632104023768692069085125, −8.153518513887687755753002447629, −7.75542122949507699534015337561, −7.29555338852647573165789085839, −6.67651918697707293120022713533, −6.52700820152537373034893095128, −6.12211488576219172631154649251, −5.43778116358633702680744602733, −5.04778809685292755127277083359, −4.96629606893722603634935763330, −4.27112550085034187750845063547, −4.11271534635929546705211120193, −3.38891401383221457119677660786, −2.92118403382524317181678753742, −2.24411964701753057802777684427, −2.14706211275372369188052938703, −1.21782147781835079564142420615, −0.17757336305304888764839935047,
0.17757336305304888764839935047, 1.21782147781835079564142420615, 2.14706211275372369188052938703, 2.24411964701753057802777684427, 2.92118403382524317181678753742, 3.38891401383221457119677660786, 4.11271534635929546705211120193, 4.27112550085034187750845063547, 4.96629606893722603634935763330, 5.04778809685292755127277083359, 5.43778116358633702680744602733, 6.12211488576219172631154649251, 6.52700820152537373034893095128, 6.67651918697707293120022713533, 7.29555338852647573165789085839, 7.75542122949507699534015337561, 8.153518513887687755753002447629, 8.178980632104023768692069085125, 8.728233298460355061167773961378, 9.014619622579347709460126090766