| L(s) = 1 | − 4·7-s + 2·9-s − 12·17-s − 12·23-s − 25-s − 8·31-s − 12·41-s − 12·47-s − 2·49-s − 8·63-s + 24·71-s − 4·73-s + 16·79-s − 5·81-s + 12·89-s + 4·97-s − 28·103-s − 12·113-s + 48·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2/3·9-s − 2.91·17-s − 2.50·23-s − 1/5·25-s − 1.43·31-s − 1.87·41-s − 1.75·47-s − 2/7·49-s − 1.00·63-s + 2.84·71-s − 0.468·73-s + 1.80·79-s − 5/9·81-s + 1.27·89-s + 0.406·97-s − 2.75·103-s − 1.12·113-s + 4.40·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.539561210487403574675283344717, −9.157322000026837505816158343106, −8.720780437434469962417611550224, −8.230129938830794240392666925692, −7.947734946671294662176519651630, −7.33782374831349734915177782979, −6.76688402776421694590616068181, −6.55439813991020394176486152580, −6.38643347295196248226107769990, −5.89685145504248507853114839610, −5.00990142224555417387955616255, −4.92788450814285464970946220317, −4.08751456399959515893395846470, −3.79555062207260394056854598536, −3.48532229815160737435781499688, −2.61865819129094450462582354274, −2.05326687888168253012791912667, −1.71445765527751321897892714700, 0, 0,
1.71445765527751321897892714700, 2.05326687888168253012791912667, 2.61865819129094450462582354274, 3.48532229815160737435781499688, 3.79555062207260394056854598536, 4.08751456399959515893395846470, 4.92788450814285464970946220317, 5.00990142224555417387955616255, 5.89685145504248507853114839610, 6.38643347295196248226107769990, 6.55439813991020394176486152580, 6.76688402776421694590616068181, 7.33782374831349734915177782979, 7.947734946671294662176519651630, 8.230129938830794240392666925692, 8.720780437434469962417611550224, 9.157322000026837505816158343106, 9.539561210487403574675283344717
