| L(s) = 1 | − 2-s − 4-s + 2·5-s + 4·7-s + 3·8-s − 2·9-s − 2·10-s − 4·14-s − 16-s + 2·18-s − 12·19-s − 2·20-s − 7·25-s − 4·28-s − 5·32-s + 8·35-s + 2·36-s − 6·37-s + 12·38-s + 6·40-s − 4·45-s − 2·49-s + 7·50-s + 18·53-s + 12·56-s − 8·63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s + 1.06·8-s − 2/3·9-s − 0.632·10-s − 1.06·14-s − 1/4·16-s + 0.471·18-s − 2.75·19-s − 0.447·20-s − 7/5·25-s − 0.755·28-s − 0.883·32-s + 1.35·35-s + 1/3·36-s − 0.986·37-s + 1.94·38-s + 0.948·40-s − 0.596·45-s − 2/7·49-s + 0.989·50-s + 2.47·53-s + 1.60·56-s − 1.00·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−8.593179580013735766728633582532, −8.264856530445176716222542589828, −8.258248518859278438486553214420, −7.49323810887805876503518242868, −6.88693062711107630940489538145, −6.37697744421228411758469042116, −5.64659770280368303942371761613, −5.44861874763839175210229504778, −4.78420083160118801963993430192, −4.16616987865897129931233797172, −3.89751875979123562820900427499, −2.57511517090571223097469756394, −1.98218366620434510274683932709, −1.52544397358253614737991046868, 0,
1.52544397358253614737991046868, 1.98218366620434510274683932709, 2.57511517090571223097469756394, 3.89751875979123562820900427499, 4.16616987865897129931233797172, 4.78420083160118801963993430192, 5.44861874763839175210229504778, 5.64659770280368303942371761613, 6.37697744421228411758469042116, 6.88693062711107630940489538145, 7.49323810887805876503518242868, 8.258248518859278438486553214420, 8.264856530445176716222542589828, 8.593179580013735766728633582532
