| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 13-s + 17-s + 19-s − 21-s + 27-s + 29-s − 31-s + 33-s + 37-s − 39-s + 41-s − 43-s + 47-s + 49-s + 51-s + 53-s + 57-s − 59-s − 61-s − 63-s − 67-s − 71-s − 73-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 13-s + 17-s + 19-s − 21-s + 27-s + 29-s − 31-s + 33-s + 37-s − 39-s + 41-s − 43-s + 47-s + 49-s + 51-s + 53-s + 57-s − 59-s − 61-s − 63-s − 67-s − 71-s − 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.861740127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.861740127\) |
| \(L(1)\) |
\(\approx\) |
\(1.439707935\) |
| \(L(1)\) |
\(\approx\) |
\(1.439707935\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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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
−24.10388537027576083895345786343, −23.0069091487347759781000647802, −22.07784971493750430081054147900, −21.46534523331561878736345627793, −20.18373050419896057437988299032, −19.758051525697163581546893455602, −19.02270584648457025552486592932, −18.11639966955166326626303867840, −16.81509588997834326476632257129, −16.175213280046622381066164327333, −15.09109201822552760623560601307, −14.364954506262933947590458268246, −13.59821989430710650386358184015, −12.547733592308642472667495635523, −11.91098684513894083356940517215, −10.309809887286241600126832643093, −9.57877843279000381979592387523, −8.98755388400476158931726735896, −7.6732272973712880716607935599, −7.02226502543883292857440293471, −5.85068551058276233553905330572, −4.444981334454629396479133269568, −3.41925686594374935941400269476, −2.66239205023711789505962934279, −1.22799839948695542960504837242,
1.22799839948695542960504837242, 2.66239205023711789505962934279, 3.41925686594374935941400269476, 4.444981334454629396479133269568, 5.85068551058276233553905330572, 7.02226502543883292857440293471, 7.6732272973712880716607935599, 8.98755388400476158931726735896, 9.57877843279000381979592387523, 10.309809887286241600126832643093, 11.91098684513894083356940517215, 12.547733592308642472667495635523, 13.59821989430710650386358184015, 14.364954506262933947590458268246, 15.09109201822552760623560601307, 16.175213280046622381066164327333, 16.81509588997834326476632257129, 18.11639966955166326626303867840, 19.02270584648457025552486592932, 19.758051525697163581546893455602, 20.18373050419896057437988299032, 21.46534523331561878736345627793, 22.07784971493750430081054147900, 23.0069091487347759781000647802, 24.10388537027576083895345786343
