| L(s) = 1 | − 3-s + 9-s + 11-s − 2·13-s − 2·17-s − 27-s − 2·29-s + 4·31-s − 33-s − 6·37-s + 2·39-s − 6·41-s − 12·43-s + 4·47-s + 2·51-s − 6·53-s + 4·59-s + 10·61-s + 4·67-s + 8·71-s − 2·73-s + 81-s − 16·83-s + 2·87-s + 6·89-s − 4·93-s − 6·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.485·17-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.174·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s + 0.583·47-s + 0.280·51-s − 0.824·53-s + 0.520·59-s + 1.28·61-s + 0.488·67-s + 0.949·71-s − 0.234·73-s + 1/9·81-s − 1.75·83-s + 0.214·87-s + 0.635·89-s − 0.414·93-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.87271614083151, −12.29755663448500, −11.89766957504998, −11.52398466193155, −11.11903483929478, −10.58455081018634, −10.01178445801066, −9.854554921083573, −9.257446671026033, −8.605077900526506, −8.354313344336566, −7.758133587575256, −7.064688397957064, −6.818579094954310, −6.439036589107633, −5.726264457284209, −5.310086225844110, −4.843451365389975, −4.385537057938573, −3.743227834487525, −3.291988497500363, −2.584453096371655, −1.947772002032142, −1.473382231048823, −0.6439446946514768, 0,
0.6439446946514768, 1.473382231048823, 1.947772002032142, 2.584453096371655, 3.291988497500363, 3.743227834487525, 4.385537057938573, 4.843451365389975, 5.310086225844110, 5.726264457284209, 6.439036589107633, 6.818579094954310, 7.064688397957064, 7.758133587575256, 8.354313344336566, 8.605077900526506, 9.257446671026033, 9.854554921083573, 10.01178445801066, 10.58455081018634, 11.11903483929478, 11.52398466193155, 11.89766957504998, 12.29755663448500, 12.87271614083151
