| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s + 13-s + 15-s − 17-s − 19-s − 21-s − 23-s + 25-s − 27-s + 29-s − 31-s + 33-s − 35-s − 37-s − 39-s + 43-s − 45-s + 47-s + 49-s + 51-s + 53-s + 55-s + 57-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s + 13-s + 15-s − 17-s − 19-s − 21-s − 23-s + 25-s − 27-s + 29-s − 31-s + 33-s − 35-s − 37-s − 39-s + 43-s − 45-s + 47-s + 49-s + 51-s + 53-s + 55-s + 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9222377102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9222377102\) |
| \(L(1)\) |
\(\approx\) |
\(0.6938617420\) |
| \(L(1)\) |
\(\approx\) |
\(0.6938617420\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 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 | \( 1 - T \) |
| 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
−24.39728687318634225290160367533, −23.769342085448817769441332289, −23.31343971374087124911881019238, −22.29244683748628722934631824279, −21.27833801296246896974919669175, −20.55646868248078872510082385480, −19.36967219063061251075202197758, −18.27205335389807611547772813975, −17.87016643385270882099361877596, −16.66767630824168238845714942897, −15.7355181169373533199178371773, −15.25594393005592266339856913512, −13.82321634769526843078901029543, −12.70579914333292207381443863449, −11.876809866433107452954965486536, −10.86836040682847504910734917577, −10.62563967916730599129247869666, −8.741166727248704071803744129079, −7.92472395261547983805697146098, −6.90798300991365509229491241153, −5.73252580608525942001011026163, −4.667136641897646057112212700789, −3.92059862075339577995648354590, −2.06839758437722211342985034565, −0.59096478641256959031163148737,
0.59096478641256959031163148737, 2.06839758437722211342985034565, 3.92059862075339577995648354590, 4.667136641897646057112212700789, 5.73252580608525942001011026163, 6.90798300991365509229491241153, 7.92472395261547983805697146098, 8.741166727248704071803744129079, 10.62563967916730599129247869666, 10.86836040682847504910734917577, 11.876809866433107452954965486536, 12.70579914333292207381443863449, 13.82321634769526843078901029543, 15.25594393005592266339856913512, 15.7355181169373533199178371773, 16.66767630824168238845714942897, 17.87016643385270882099361877596, 18.27205335389807611547772813975, 19.36967219063061251075202197758, 20.55646868248078872510082385480, 21.27833801296246896974919669175, 22.29244683748628722934631824279, 23.31343971374087124911881019238, 23.769342085448817769441332289, 24.39728687318634225290160367533
