| L(s) = 1 | + 2-s + 4-s − 9·7-s + 8-s − 5·9-s − 9·14-s + 16-s − 3·17-s − 5·18-s + 23-s + 7·25-s − 9·28-s + 3·31-s + 32-s − 3·34-s − 5·36-s − 6·41-s + 46-s − 6·47-s + 47·49-s + 7·50-s − 9·56-s + 3·62-s + 45·63-s + 64-s − 3·68-s − 24·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 3.40·7-s + 0.353·8-s − 5/3·9-s − 2.40·14-s + 1/4·16-s − 0.727·17-s − 1.17·18-s + 0.208·23-s + 7/5·25-s − 1.70·28-s + 0.538·31-s + 0.176·32-s − 0.514·34-s − 5/6·36-s − 0.937·41-s + 0.147·46-s − 0.875·47-s + 47/7·49-s + 0.989·50-s − 1.20·56-s + 0.381·62-s + 5.66·63-s + 1/8·64-s − 0.363·68-s − 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 193 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 14 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 87 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−10.43080020248064986585118434914, −9.891718230555023862651161536501, −9.438977706266847773294700566750, −8.704000099970979670974486573773, −8.647771816880424792768985371047, −7.38833426728406096214604755120, −6.75617877570598088183816301431, −6.49818756310180090266999927656, −5.96643657379659688078199079593, −5.49031665363328806426308509722, −4.47291422840232191202006040667, −3.52641841993105450729084438810, −2.93021005274317878280441630391, −2.80297748780335495293888795998, 0,
2.80297748780335495293888795998, 2.93021005274317878280441630391, 3.52641841993105450729084438810, 4.47291422840232191202006040667, 5.49031665363328806426308509722, 5.96643657379659688078199079593, 6.49818756310180090266999927656, 6.75617877570598088183816301431, 7.38833426728406096214604755120, 8.647771816880424792768985371047, 8.704000099970979670974486573773, 9.438977706266847773294700566750, 9.891718230555023862651161536501, 10.43080020248064986585118434914
