| L(s) = 1 | + 2·3-s − 7-s + 3·9-s − 8·19-s − 2·21-s − 6·25-s + 4·27-s − 20·29-s + 16·31-s + 12·37-s − 16·47-s + 49-s − 20·53-s − 16·57-s − 24·59-s − 3·63-s − 12·75-s + 5·81-s − 24·83-s − 40·87-s + 32·93-s − 4·109-s + 24·111-s + 36·113-s − 22·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 9-s − 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.769·27-s − 3.71·29-s + 2.87·31-s + 1.97·37-s − 2.33·47-s + 1/7·49-s − 2.74·53-s − 2.11·57-s − 3.12·59-s − 0.377·63-s − 1.38·75-s + 5/9·81-s − 2.63·83-s − 4.28·87-s + 3.31·93-s − 0.383·109-s + 2.27·111-s + 3.38·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.289231676437400116460418770344, −7.917190719920927542416951015280, −7.84194967359953188564623556072, −7.13572102783324715252694770569, −6.54428519689900750892915035611, −6.05800461819552664757143425288, −5.90015243373728427076887013140, −4.81364713899235799057960760690, −4.35622436888448856514855173618, −4.09990139504038953260193041491, −3.10934300001651305626809863177, −3.06416831077517725944120442154, −1.98451470429642242976214434683, −1.70572377211895013061556457420, 0,
1.70572377211895013061556457420, 1.98451470429642242976214434683, 3.06416831077517725944120442154, 3.10934300001651305626809863177, 4.09990139504038953260193041491, 4.35622436888448856514855173618, 4.81364713899235799057960760690, 5.90015243373728427076887013140, 6.05800461819552664757143425288, 6.54428519689900750892915035611, 7.13572102783324715252694770569, 7.84194967359953188564623556072, 7.917190719920927542416951015280, 8.289231676437400116460418770344
