L(s) = 1 | + 3-s + 9-s − 2·11-s + 2·13-s − 4·17-s + 8·23-s + 27-s − 2·31-s − 2·33-s − 8·37-s + 2·39-s + 2·41-s − 2·43-s − 10·47-s − 4·51-s + 2·53-s + 4·59-s + 10·61-s + 2·67-s + 8·69-s + 12·71-s + 10·73-s − 16·79-s + 81-s − 16·83-s − 14·89-s − 2·93-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.970·17-s + 1.66·23-s + 0.192·27-s − 0.359·31-s − 0.348·33-s − 1.31·37-s + 0.320·39-s + 0.312·41-s − 0.304·43-s − 1.45·47-s − 0.560·51-s + 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.244·67-s + 0.963·69-s + 1.42·71-s + 1.17·73-s − 1.80·79-s + 1/9·81-s − 1.75·83-s − 1.48·89-s − 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.587254788\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.587254788\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.19817193892655, −13.95063655649613, −13.17581254040843, −12.90947419531165, −12.63873116997995, −11.63521907024192, −11.20198287596510, −10.91642590742962, −10.02214142364829, −9.900250566666797, −8.911680618812477, −8.728685755117431, −8.288169759633086, −7.491508221035230, −7.028705215549728, −6.597530379944704, −5.855634191377192, −5.091147218897041, −4.813776581871488, −3.922980662755433, −3.425052566832367, −2.789387547524379, −2.144942332165920, −1.448106238590092, −0.5292822448578361,
0.5292822448578361, 1.448106238590092, 2.144942332165920, 2.789387547524379, 3.425052566832367, 3.922980662755433, 4.813776581871488, 5.091147218897041, 5.855634191377192, 6.597530379944704, 7.028705215549728, 7.491508221035230, 8.288169759633086, 8.728685755117431, 8.911680618812477, 9.900250566666797, 10.02214142364829, 10.91642590742962, 11.20198287596510, 11.63521907024192, 12.63873116997995, 12.90947419531165, 13.17581254040843, 13.95063655649613, 14.19817193892655