| L(s) = 1 | − 7-s + 4·13-s − 4·17-s − 4·19-s + 8·23-s − 2·29-s + 8·31-s − 8·37-s − 6·41-s − 8·43-s + 8·47-s + 49-s − 4·59-s − 6·61-s − 8·67-s + 12·71-s − 4·73-s + 4·79-s + 10·89-s − 4·91-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 4·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.10·13-s − 0.970·17-s − 0.917·19-s + 1.66·23-s − 0.371·29-s + 1.43·31-s − 1.31·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.520·59-s − 0.768·61-s − 0.977·67-s + 1.42·71-s − 0.468·73-s + 0.450·79-s + 1.05·89-s − 0.419·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.366·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.843493347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.843493347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.35789699590610, −15.13267757464277, −14.13743790354995, −13.68828778646265, −13.22000280798258, −12.82303228956959, −12.11726154549860, −11.54826631753258, −10.93924004390107, −10.54900920442563, −9.998927964937771, −9.121228279118007, −8.748597132638859, −8.396659766396650, −7.486727489938140, −6.821794408791267, −6.432079913696271, −5.865298413992079, −4.939373378224022, −4.549457565423642, −3.625752594641684, −3.181727669609237, −2.289788481823478, −1.508560824930619, −0.5455675932652253,
0.5455675932652253, 1.508560824930619, 2.289788481823478, 3.181727669609237, 3.625752594641684, 4.549457565423642, 4.939373378224022, 5.865298413992079, 6.432079913696271, 6.821794408791267, 7.486727489938140, 8.396659766396650, 8.748597132638859, 9.121228279118007, 9.998927964937771, 10.54900920442563, 10.93924004390107, 11.54826631753258, 12.11726154549860, 12.82303228956959, 13.22000280798258, 13.68828778646265, 14.13743790354995, 15.13267757464277, 15.35789699590610
