| L(s) = 1 | − 4·7-s − 2·13-s − 16·19-s + 5·25-s − 4·31-s − 20·37-s + 8·43-s + 7·49-s − 14·61-s − 16·67-s − 20·73-s − 4·79-s + 8·91-s − 14·97-s + 20·103-s + 4·109-s + 11·121-s + 127-s + 131-s + 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.554·13-s − 3.67·19-s + 25-s − 0.718·31-s − 3.28·37-s + 1.21·43-s + 49-s − 1.79·61-s − 1.95·67-s − 2.34·73-s − 0.450·79-s + 0.838·91-s − 1.42·97-s + 1.97·103-s + 0.383·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−9.211713298004280934576305971546, −8.990890859707656633183436792477, −8.802252600257081365423287630570, −8.483252289245808846100882759500, −7.73338015764669673333066923383, −7.33345691721823336896265067753, −6.86841691726000487303562102600, −6.63272656572301314153677296946, −6.03614083244652445333918166232, −6.02312489529303817182564551620, −5.17631427365747618133931549080, −4.75215650835674560577797473830, −4.06553442717947310899147176976, −4.00417543536580328788535595394, −3.06771075596467426674622927791, −2.89245730339382694666960367400, −2.09217505903120938420458680056, −1.61736177436991935904221204202, 0, 0,
1.61736177436991935904221204202, 2.09217505903120938420458680056, 2.89245730339382694666960367400, 3.06771075596467426674622927791, 4.00417543536580328788535595394, 4.06553442717947310899147176976, 4.75215650835674560577797473830, 5.17631427365747618133931549080, 6.02312489529303817182564551620, 6.03614083244652445333918166232, 6.63272656572301314153677296946, 6.86841691726000487303562102600, 7.33345691721823336896265067753, 7.73338015764669673333066923383, 8.483252289245808846100882759500, 8.802252600257081365423287630570, 8.990890859707656633183436792477, 9.211713298004280934576305971546
