| L(s) = 1 | − 3·3-s + 5-s + 6·9-s + 11-s + 4·13-s − 3·15-s − 2·17-s − 6·19-s + 5·23-s − 4·25-s − 9·27-s + 10·29-s + 31-s − 3·33-s − 5·37-s − 12·39-s + 2·41-s + 8·43-s + 6·45-s + 8·47-s + 6·51-s − 6·53-s + 55-s + 18·57-s + 3·59-s + 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s + 0.301·11-s + 1.10·13-s − 0.774·15-s − 0.485·17-s − 1.37·19-s + 1.04·23-s − 4/5·25-s − 1.73·27-s + 1.85·29-s + 0.179·31-s − 0.522·33-s − 0.821·37-s − 1.92·39-s + 0.312·41-s + 1.21·43-s + 0.894·45-s + 1.16·47-s + 0.840·51-s − 0.824·53-s + 0.134·55-s + 2.38·57-s + 0.390·59-s + 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.209403676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209403676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.54835773065274067173057730543, −6.72496156498857491892507194459, −6.28504064266312100949805459883, −5.91492408265913949324599251189, −5.04813557951775111822692683649, −4.45791528491933496347678187080, −3.76832108585204209715568685401, −2.47568663901626477311986587417, −1.43849650902406242082522223739, −0.63877537931902655978837422699,
0.63877537931902655978837422699, 1.43849650902406242082522223739, 2.47568663901626477311986587417, 3.76832108585204209715568685401, 4.45791528491933496347678187080, 5.04813557951775111822692683649, 5.91492408265913949324599251189, 6.28504064266312100949805459883, 6.72496156498857491892507194459, 7.54835773065274067173057730543
