L(s) = 1 | + (0.866 − 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s − i·27-s + 29-s + (−0.5 − 0.866i)31-s + (−0.866 − 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s − 41-s − i·43-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s − i·27-s + 29-s + (−0.5 − 0.866i)31-s + (−0.866 − 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s − 41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5508495101 - 1.556425609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5508495101 - 1.556425609i\) |
\(L(1)\) |
\(\approx\) |
\(1.114803552 - 0.4822781678i\) |
\(L(1)\) |
\(\approx\) |
\(1.114803552 - 0.4822781678i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 - iT \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.89917375519856039506759673317, −25.00996890542416849444676354006, −24.0137149821975039738835099450, −23.066373575739304031488055773751, −21.780978405764138119687601068245, −21.34783958047517607448391250337, −20.10745754674758980486028121299, −19.71004023614532179386814111700, −18.486142283184810406911468975010, −17.58137834338805284294934430148, −16.24756962527978145933799492472, −15.5957251786539563644927117889, −14.64810189373547782526120965049, −13.75274920238567865553730674935, −12.89153729214914493550191478458, −11.59342411027073447611029651454, −10.482517570697316569290367472, −9.51937307388044753325978146623, −8.7466921908754274286855559452, −7.598362621421491656962540331, −6.627822889087063794229710926533, −4.896197741510242668892740344103, −4.22411058318721066964624129525, −2.79362012237361480003593515889, −1.84011330626874518932297031160,
0.41436846916003780654128931825, 1.95290919458244534062908280059, 3.04448685912404642789711656076, 4.11955803711179041878400958572, 5.726541193141967032007712903721, 6.71141362491531381714580636208, 8.155961717035091958308996808785, 8.36742116209516728560209099308, 9.83239793709113105070853860732, 10.75911488180791702285367403323, 12.125809277393788702301251235251, 13.05565161897819355637681184976, 13.74687655495938705144354846040, 14.83409188188457317045697245964, 15.58185249301214445807606308824, 16.75299564713503196090648713781, 18.00173824363871870950842279796, 18.62216065304109617473401852968, 19.652336875806843959891913873907, 20.348011394557313957794043427106, 21.285295201946106265241957663242, 22.25410235566173433845720942143, 23.4934040720527903580907011659, 24.21615602064077802178779652070, 25.05519507354105617364879193958