| L(s) = 1 | − 5-s − 11-s + 6·17-s + 4·19-s − 8·23-s + 25-s − 4·29-s + 4·31-s − 6·37-s − 6·41-s − 4·43-s + 4·47-s − 12·53-s + 55-s + 8·59-s + 4·61-s − 8·67-s + 12·71-s + 8·73-s − 8·79-s + 12·83-s − 6·85-s + 10·89-s − 4·95-s + 12·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.742·29-s + 0.718·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 1.64·53-s + 0.134·55-s + 1.04·59-s + 0.512·61-s − 0.977·67-s + 1.42·71-s + 0.936·73-s − 0.900·79-s + 1.31·83-s − 0.650·85-s + 1.05·89-s − 0.410·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.04378664077861, −13.65098567493974, −13.04037963925310, −12.43149457393382, −12.00091390296622, −11.76485572116846, −11.16461861659480, −10.44719436132473, −10.12640564953854, −9.653664120010598, −9.144994939579195, −8.284976772086320, −8.100172090931717, −7.560060144020053, −7.094579942450447, −6.363985793137320, −5.888770289993279, −5.182064179866905, −4.961407971383299, −4.004304643891872, −3.548556773613994, −3.146525558238922, −2.268134204777559, −1.624221619107739, −0.8411181709335724, 0,
0.8411181709335724, 1.624221619107739, 2.268134204777559, 3.146525558238922, 3.548556773613994, 4.004304643891872, 4.961407971383299, 5.182064179866905, 5.888770289993279, 6.363985793137320, 7.094579942450447, 7.560060144020053, 8.100172090931717, 8.284976772086320, 9.144994939579195, 9.653664120010598, 10.12640564953854, 10.44719436132473, 11.16461861659480, 11.76485572116846, 12.00091390296622, 12.43149457393382, 13.04037963925310, 13.65098567493974, 14.04378664077861
