| L(s) = 1 | − 6·9-s − 4·16-s − 8·17-s − 6·23-s + 25-s − 10·29-s + 2·43-s − 49-s − 18·53-s − 20·61-s + 6·79-s + 27·81-s + 28·101-s + 8·103-s − 8·107-s − 6·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·9-s − 16-s − 1.94·17-s − 1.25·23-s + 1/5·25-s − 1.85·29-s + 0.304·43-s − 1/7·49-s − 2.47·53-s − 2.56·61-s + 0.675·79-s + 3·81-s + 2.78·101-s + 0.788·103-s − 0.773·107-s − 0.564·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + 0.0813·151-s + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.342165744759600001177971385751, −9.216434654102498881900676986568, −8.647665992310408386362152389604, −8.605054182995359466452401331806, −7.79564503359671757395302267518, −7.71968146717104930633286583149, −7.08747754694164583198811242244, −6.43564563033490143101557607390, −6.14853283728876422680485728492, −6.00146053406771464552867517460, −5.29015450905288561683676006858, −4.73171856981058939699108533044, −4.54102261865655065215199450947, −3.70358442199176084348655640008, −3.38289976035022851334095946634, −2.58707040853750830679738532681, −2.28058876041277896516478184491, −1.68179844572142180891700624983, 0, 0,
1.68179844572142180891700624983, 2.28058876041277896516478184491, 2.58707040853750830679738532681, 3.38289976035022851334095946634, 3.70358442199176084348655640008, 4.54102261865655065215199450947, 4.73171856981058939699108533044, 5.29015450905288561683676006858, 6.00146053406771464552867517460, 6.14853283728876422680485728492, 6.43564563033490143101557607390, 7.08747754694164583198811242244, 7.71968146717104930633286583149, 7.79564503359671757395302267518, 8.605054182995359466452401331806, 8.647665992310408386362152389604, 9.216434654102498881900676986568, 9.342165744759600001177971385751
