| L(s) = 1 | − 5-s − 4·11-s − 13-s + 4·17-s + 2·23-s + 25-s + 29-s − 4·31-s + 2·37-s − 4·43-s − 7·49-s + 2·53-s + 4·55-s − 2·59-s + 10·61-s + 65-s − 4·67-s − 2·71-s + 10·73-s − 4·83-s − 4·85-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.277·13-s + 0.970·17-s + 0.417·23-s + 1/5·25-s + 0.185·29-s − 0.718·31-s + 0.328·37-s − 0.609·43-s − 49-s + 0.274·53-s + 0.539·55-s − 0.260·59-s + 1.28·61-s + 0.124·65-s − 0.488·67-s − 0.237·71-s + 1.17·73-s − 0.439·83-s − 0.433·85-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.421183797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421183797\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.79614789091649, −12.45470634132749, −11.75725146504653, −11.46786339591008, −10.93348845543684, −10.49485941751994, −9.954915211682481, −9.735073831111007, −9.004972711940467, −8.531173017451366, −8.043281183992477, −7.673102454151222, −7.235839946523611, −6.755688386262769, −6.074704678032103, −5.581116490567086, −5.070009354211741, −4.755506152612528, −4.040641585530823, −3.418032099284647, −3.057096097014534, −2.427319593433134, −1.825371913232034, −1.032558731897231, −0.3568815053024447,
0.3568815053024447, 1.032558731897231, 1.825371913232034, 2.427319593433134, 3.057096097014534, 3.418032099284647, 4.040641585530823, 4.755506152612528, 5.070009354211741, 5.581116490567086, 6.074704678032103, 6.755688386262769, 7.235839946523611, 7.673102454151222, 8.043281183992477, 8.531173017451366, 9.004972711940467, 9.735073831111007, 9.954915211682481, 10.49485941751994, 10.93348845543684, 11.46786339591008, 11.75725146504653, 12.45470634132749, 12.79614789091649
