L(s) = 1 | − 1.18·3-s + 2.76·7-s − 1.59·9-s − 1.40·11-s − 3.57·13-s − 1.66·17-s − 2.85·19-s − 3.28·21-s − 3.04·23-s + 5.45·27-s + 8.24·29-s − 31-s + 1.66·33-s + 8.45·37-s + 4.24·39-s + 6.20·41-s + 6.55·43-s + 4.20·47-s + 0.665·49-s + 1.97·51-s + 9.42·53-s + 3.38·57-s − 12.9·59-s − 13.0·61-s − 4.40·63-s + 9.44·67-s + 3.60·69-s + ⋯ |
L(s) = 1 | − 0.685·3-s + 1.04·7-s − 0.530·9-s − 0.423·11-s − 0.991·13-s − 0.404·17-s − 0.654·19-s − 0.717·21-s − 0.633·23-s + 1.04·27-s + 1.53·29-s − 0.179·31-s + 0.289·33-s + 1.39·37-s + 0.679·39-s + 0.968·41-s + 0.999·43-s + 0.613·47-s + 0.0951·49-s + 0.276·51-s + 1.29·53-s + 0.448·57-s − 1.68·59-s − 1.66·61-s − 0.554·63-s + 1.15·67-s + 0.434·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.197436748\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.197436748\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 1.18T + 3T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 + 3.57T + 13T^{2} \) |
| 17 | \( 1 + 1.66T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 + 3.04T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 37 | \( 1 - 8.45T + 37T^{2} \) |
| 41 | \( 1 - 6.20T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 - 4.20T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 + 3.82T + 71T^{2} \) |
| 73 | \( 1 + 7.81T + 73T^{2} \) |
| 79 | \( 1 - 0.515T + 79T^{2} \) |
| 83 | \( 1 - 5.13T + 83T^{2} \) |
| 89 | \( 1 - 5.03T + 89T^{2} \) |
| 97 | \( 1 - 5.28T + 97T^{2} \) |
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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
−8.596597831198964086040687603768, −7.932671885229411679799455021731, −7.28953147618432492789931016308, −6.22981917893499137120227277349, −5.71977886205306456584307149054, −4.69492916386897965554201785450, −4.44591876284515975278254453244, −2.86541683397273441302976990202, −2.11081558921774354381346922605, −0.66954384301089082802243722020,
0.66954384301089082802243722020, 2.11081558921774354381346922605, 2.86541683397273441302976990202, 4.44591876284515975278254453244, 4.69492916386897965554201785450, 5.71977886205306456584307149054, 6.22981917893499137120227277349, 7.28953147618432492789931016308, 7.932671885229411679799455021731, 8.596597831198964086040687603768