| L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·13-s + 6·17-s − 4·19-s − 2·21-s − 8·23-s − 27-s − 4·31-s − 10·37-s + 2·39-s + 4·41-s − 2·43-s − 8·47-s − 3·49-s − 6·51-s − 6·53-s + 4·57-s − 10·59-s − 6·61-s + 2·63-s + 6·67-s + 8·69-s − 8·71-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.436·21-s − 1.66·23-s − 0.192·27-s − 0.718·31-s − 1.64·37-s + 0.320·39-s + 0.624·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s − 1.30·59-s − 0.768·61-s + 0.251·63-s + 0.733·67-s + 0.963·69-s − 0.949·71-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5139851467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5139851467\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.88714622340695, −12.47188427087030, −12.10484590318331, −11.76802079100053, −11.18576119376045, −10.68869557706612, −10.35256883406751, −9.783778579472115, −9.474267949307385, −8.700838736337756, −8.209244618670821, −7.809109897917794, −7.399311289485536, −6.801849904619792, −6.144102506978552, −5.845758169879858, −5.222303149398840, −4.773470812525506, −4.337988670474810, −3.592370453854635, −3.208185885228348, −2.246860290790305, −1.744960898103515, −1.298704534480600, −0.2063744630532592,
0.2063744630532592, 1.298704534480600, 1.744960898103515, 2.246860290790305, 3.208185885228348, 3.592370453854635, 4.337988670474810, 4.773470812525506, 5.222303149398840, 5.845758169879858, 6.144102506978552, 6.801849904619792, 7.399311289485536, 7.809109897917794, 8.209244618670821, 8.700838736337756, 9.474267949307385, 9.783778579472115, 10.35256883406751, 10.68869557706612, 11.18576119376045, 11.76802079100053, 12.10484590318331, 12.47188427087030, 12.88714622340695
