| L(s) = 1 | + 2·3-s + 9-s + 5·11-s + 8·17-s + 2·19-s + 7·23-s − 4·27-s − 3·29-s − 4·31-s + 10·33-s + 37-s + 2·41-s − 3·43-s + 6·47-s + 16·51-s − 10·53-s + 4·57-s + 4·59-s + 6·61-s − 13·67-s + 14·69-s + 5·71-s + 6·73-s − 13·79-s − 11·81-s + 16·83-s − 6·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.50·11-s + 1.94·17-s + 0.458·19-s + 1.45·23-s − 0.769·27-s − 0.557·29-s − 0.718·31-s + 1.74·33-s + 0.164·37-s + 0.312·41-s − 0.457·43-s + 0.875·47-s + 2.24·51-s − 1.37·53-s + 0.529·57-s + 0.520·59-s + 0.768·61-s − 1.58·67-s + 1.68·69-s + 0.593·71-s + 0.702·73-s − 1.46·79-s − 1.22·81-s + 1.75·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.073035651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.073035651\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.53320265025681607572724183668, −7.33521442592951872710329664996, −6.34794041356949552768820188495, −5.64046926486444984862753296319, −4.89249892555955693039583334525, −3.80948380079618921351508908616, −3.46369473923162714776849414642, −2.77979036284652674425005709692, −1.69959270403433982266792826058, −0.984085766779887252920356039055,
0.984085766779887252920356039055, 1.69959270403433982266792826058, 2.77979036284652674425005709692, 3.46369473923162714776849414642, 3.80948380079618921351508908616, 4.89249892555955693039583334525, 5.64046926486444984862753296319, 6.34794041356949552768820188495, 7.33521442592951872710329664996, 7.53320265025681607572724183668
