| L(s) = 1 | + 4·4-s + 4·7-s + 3·13-s + 12·16-s − 9·19-s + 5·25-s + 16·28-s − 37-s + 5·43-s + 9·49-s + 12·52-s + 32·64-s + 22·67-s − 27·73-s − 36·76-s − 26·79-s + 12·91-s − 24·97-s + 20·100-s − 33·103-s + 17·109-s + 48·112-s − 11·121-s + 127-s + 131-s − 36·133-s + 137-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.51·7-s + 0.832·13-s + 3·16-s − 2.06·19-s + 25-s + 3.02·28-s − 0.164·37-s + 0.762·43-s + 9/7·49-s + 1.66·52-s + 4·64-s + 2.68·67-s − 3.16·73-s − 4.12·76-s − 2.92·79-s + 1.25·91-s − 2.43·97-s + 2·100-s − 3.25·103-s + 1.62·109-s + 4.53·112-s − 121-s + 0.0887·127-s + 0.0873·131-s − 3.12·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.908467081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.908467081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.96640617341950846294003090899, −10.80160183622685390093008698612, −10.14675132633544754035340080738, −10.09037317724438320334506883848, −8.888586260839942921327067006715, −8.761570856361985341027472109431, −8.089800982392398156268615062530, −8.017048840680845066199579534820, −7.24957862861028457395485576462, −6.99457579740512571164538674499, −6.39914518115292003908587222515, −6.13944624876711242862314085014, −5.46350057918716772869735603377, −5.10101186800904647040691920912, −4.06430820569097814351182259876, −4.01687775311236423892081915788, −2.69891741342343106192034128999, −2.65823283055454640986641600539, −1.63767714876374694294915229184, −1.37815594620778404487983892563,
1.37815594620778404487983892563, 1.63767714876374694294915229184, 2.65823283055454640986641600539, 2.69891741342343106192034128999, 4.01687775311236423892081915788, 4.06430820569097814351182259876, 5.10101186800904647040691920912, 5.46350057918716772869735603377, 6.13944624876711242862314085014, 6.39914518115292003908587222515, 6.99457579740512571164538674499, 7.24957862861028457395485576462, 8.017048840680845066199579534820, 8.089800982392398156268615062530, 8.761570856361985341027472109431, 8.888586260839942921327067006715, 10.09037317724438320334506883848, 10.14675132633544754035340080738, 10.80160183622685390093008698612, 10.96640617341950846294003090899
