| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.391388339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391388339\) |
| \(L(1)\) |
\(\approx\) |
\(1.547751610\) |
| \(L(1)\) |
\(\approx\) |
\(1.547751610\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 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
−29.70619478390247684866308909024, −28.55956597869325988354822117306, −27.76095853591811345532120017403, −26.6044557166879163880011819094, −24.9997311416078033110086462497, −23.77936798536478412072227582889, −23.490048418832871368530892341360, −22.54984404651302574309166770398, −21.20624288076856778125338726191, −20.62769836871831760281793046165, −19.0049738497768123860589981497, −17.910955739023364850713895019146, −16.38845646488609529555706165004, −15.76305727282514289807591967513, −14.64853320683071550339186690912, −13.2159466464713963408494632120, −12.10005595251907718671952957330, −11.29724056641581031761690264632, −10.53168806524888172117251773731, −8.04224058167483907554101273723, −7.10044898312757032328357569770, −5.52136041664144176277019279530, −4.75061487504063370435485679808, −3.38324803749523647618934500777, −1.20479544152615396627284677538,
1.20479544152615396627284677538, 3.38324803749523647618934500777, 4.75061487504063370435485679808, 5.52136041664144176277019279530, 7.10044898312757032328357569770, 8.04224058167483907554101273723, 10.53168806524888172117251773731, 11.29724056641581031761690264632, 12.10005595251907718671952957330, 13.2159466464713963408494632120, 14.64853320683071550339186690912, 15.76305727282514289807591967513, 16.38845646488609529555706165004, 17.910955739023364850713895019146, 19.0049738497768123860589981497, 20.62769836871831760281793046165, 21.20624288076856778125338726191, 22.54984404651302574309166770398, 23.490048418832871368530892341360, 23.77936798536478412072227582889, 24.9997311416078033110086462497, 26.6044557166879163880011819094, 27.76095853591811345532120017403, 28.55956597869325988354822117306, 29.70619478390247684866308909024
