| L(s) = 1 | + 4-s − 2·5-s + 16-s + 6·17-s − 2·20-s − 6·25-s − 4·37-s − 2·41-s − 4·43-s + 16·47-s − 16·59-s + 64-s − 20·67-s + 6·68-s − 2·80-s − 24·83-s − 12·85-s + 22·89-s − 6·100-s − 10·101-s − 8·109-s + 2·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 1/4·16-s + 1.45·17-s − 0.447·20-s − 6/5·25-s − 0.657·37-s − 0.312·41-s − 0.609·43-s + 2.33·47-s − 2.08·59-s + 1/8·64-s − 2.44·67-s + 0.727·68-s − 0.223·80-s − 2.63·83-s − 1.30·85-s + 2.33·89-s − 3/5·100-s − 0.995·101-s − 0.766·109-s + 2/11·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
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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
−7.83795331822675730800068939214, −7.65250701529924164745407792491, −7.29852013360318005316060688132, −6.83691740064393344078043868522, −6.13597729610987507467809088549, −5.81164637525282757129351675335, −5.44545109790415515878140340354, −4.71966757523113302473352781857, −4.23678690928235581310043684621, −3.71232043159847799785605204996, −3.24592510915259754966428324866, −2.74000300628729820823480545755, −1.87617232443919620662054724633, −1.21442038284239623163596349199, 0,
1.21442038284239623163596349199, 1.87617232443919620662054724633, 2.74000300628729820823480545755, 3.24592510915259754966428324866, 3.71232043159847799785605204996, 4.23678690928235581310043684621, 4.71966757523113302473352781857, 5.44545109790415515878140340354, 5.81164637525282757129351675335, 6.13597729610987507467809088549, 6.83691740064393344078043868522, 7.29852013360318005316060688132, 7.65250701529924164745407792491, 7.83795331822675730800068939214
