| L(s) = 1 | + 3·2-s − 6·3-s + 2·4-s − 18·6-s − 3·8-s + 27·9-s − 12·12-s − 4·13-s − 3·16-s + 27·17-s + 81·18-s + 11·19-s + 18·24-s + 38·25-s − 12·26-s − 108·27-s + 78·29-s + 32·31-s − 12·32-s + 81·34-s + 54·36-s + 34·37-s + 33·38-s + 24·39-s + 21·41-s + 61·43-s − 84·47-s + ⋯ |
| L(s) = 1 | + 3/2·2-s − 2·3-s + 1/2·4-s − 3·6-s − 3/8·8-s + 3·9-s − 12-s − 0.307·13-s − 0.187·16-s + 1.58·17-s + 9/2·18-s + 0.578·19-s + 3/4·24-s + 1.51·25-s − 0.461·26-s − 4·27-s + 2.68·29-s + 1.03·31-s − 3/8·32-s + 2.38·34-s + 3/2·36-s + 0.918·37-s + 0.868·38-s + 8/13·39-s + 0.512·41-s + 1.41·43-s − 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.904190910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.904190910\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 38 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 239 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T - 153 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 27 T + 532 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 78 T + 2869 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 32 T + 63 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T - 213 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 21 T + 1828 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 84 T + 4561 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 87 T + 6004 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 56 T - 585 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 65 T - 1104 T^{2} - 65 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 38 T - 4797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 84 T + 9241 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 216 T + 23473 T^{2} + 216 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 115 T + 3816 T^{2} + 115 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.11669688931542037299080453647, −11.02659186376366523737657167535, −10.14237799913061813521768649407, −10.13505996471180596623718472121, −9.682704663445021831431995170022, −9.010883321488200835257538886304, −8.001688992779216487353204428386, −7.993593059364032293741358331582, −6.89342039273959946186132681787, −6.87968109070623308734591693521, −6.26418019435859982407687098239, −5.61101856944909265928583265132, −5.40495983897568331095246361414, −4.90636861967000180375090376315, −4.49270493950554990215197436261, −4.14407039002768043842349032770, −3.32176410168306886314795293215, −2.65694503339670201368950962900, −1.16480833981588281035309568746, −0.811001450255143075698720187456,
0.811001450255143075698720187456, 1.16480833981588281035309568746, 2.65694503339670201368950962900, 3.32176410168306886314795293215, 4.14407039002768043842349032770, 4.49270493950554990215197436261, 4.90636861967000180375090376315, 5.40495983897568331095246361414, 5.61101856944909265928583265132, 6.26418019435859982407687098239, 6.87968109070623308734591693521, 6.89342039273959946186132681787, 7.993593059364032293741358331582, 8.001688992779216487353204428386, 9.010883321488200835257538886304, 9.682704663445021831431995170022, 10.13505996471180596623718472121, 10.14237799913061813521768649407, 11.02659186376366523737657167535, 11.11669688931542037299080453647
