L(s) = 1 | + 2·3-s + 4·5-s + 3·9-s + 8·15-s + 2·17-s − 2·19-s + 8·23-s + 5·25-s + 10·27-s + 4·29-s + 4·31-s + 6·37-s − 4·41-s − 16·43-s + 12·45-s − 4·47-s + 4·51-s + 10·53-s − 4·57-s + 6·59-s − 4·61-s − 12·67-s + 16·69-s + 14·73-s + 10·75-s − 8·79-s + 20·81-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.78·5-s + 9-s + 2.06·15-s + 0.485·17-s − 0.458·19-s + 1.66·23-s + 25-s + 1.92·27-s + 0.742·29-s + 0.718·31-s + 0.986·37-s − 0.624·41-s − 2.43·43-s + 1.78·45-s − 0.583·47-s + 0.560·51-s + 1.37·53-s − 0.529·57-s + 0.781·59-s − 0.512·61-s − 1.46·67-s + 1.92·69-s + 1.63·73-s + 1.15·75-s − 0.900·79-s + 20/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.876525938\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.876525938\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.18757079550393296912608702903, −10.07146420874460009865246895887, −9.614538880047030733502274423082, −9.385629454929749075372839167549, −8.703615828575611585298132826406, −8.474705804600539351787323355993, −8.234528934786063440154860140464, −7.48875275460665815851260761246, −6.92599685347346531152144476027, −6.64662886532235588787612337134, −6.25629738510967164543363407084, −5.59595534081014243340137400865, −5.12459784446540046217163211228, −4.74651553921977127490508082889, −4.08022324019067241610113110243, −3.34644279965749150319468135345, −2.68402778144868303550679469172, −2.61132341614591274320935923971, −1.62418597665957358110667484267, −1.22054487758578256379095422854,
1.22054487758578256379095422854, 1.62418597665957358110667484267, 2.61132341614591274320935923971, 2.68402778144868303550679469172, 3.34644279965749150319468135345, 4.08022324019067241610113110243, 4.74651553921977127490508082889, 5.12459784446540046217163211228, 5.59595534081014243340137400865, 6.25629738510967164543363407084, 6.64662886532235588787612337134, 6.92599685347346531152144476027, 7.48875275460665815851260761246, 8.234528934786063440154860140464, 8.474705804600539351787323355993, 8.703615828575611585298132826406, 9.385629454929749075372839167549, 9.614538880047030733502274423082, 10.07146420874460009865246895887, 10.18757079550393296912608702903