| L(s) = 1 | − 2·7-s − 4·11-s − 13-s − 17-s + 4·19-s − 4·29-s + 6·31-s − 2·37-s + 8·41-s − 8·43-s − 8·47-s − 3·49-s + 6·53-s − 12·59-s + 4·61-s − 4·67-s + 2·71-s + 16·73-s + 8·77-s − 4·79-s + 12·83-s + 10·89-s + 2·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s − 0.277·13-s − 0.242·17-s + 0.917·19-s − 0.742·29-s + 1.07·31-s − 0.328·37-s + 1.24·41-s − 1.21·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s − 1.56·59-s + 0.512·61-s − 0.488·67-s + 0.237·71-s + 1.87·73-s + 0.911·77-s − 0.450·79-s + 1.31·83-s + 1.05·89-s + 0.209·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363848821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363848821\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.50213133630450, −12.01450367011032, −11.49280433212170, −11.16840888548314, −10.51908086027739, −10.15238283092469, −9.830525165674055, −9.274742059011357, −8.962035914398335, −8.204506646443126, −7.889614769117805, −7.463696705497906, −6.969781105782554, −6.337073437887657, −6.105096199531993, −5.378543090106834, −4.976888193120406, −4.635212687354963, −3.819727675575761, −3.286431144700408, −2.987194075473805, −2.302699364184107, −1.853730143606738, −0.9319492752852435, −0.3436045185856677,
0.3436045185856677, 0.9319492752852435, 1.853730143606738, 2.302699364184107, 2.987194075473805, 3.286431144700408, 3.819727675575761, 4.635212687354963, 4.976888193120406, 5.378543090106834, 6.105096199531993, 6.337073437887657, 6.969781105782554, 7.463696705497906, 7.889614769117805, 8.204506646443126, 8.962035914398335, 9.274742059011357, 9.830525165674055, 10.15238283092469, 10.51908086027739, 11.16840888548314, 11.49280433212170, 12.01450367011032, 12.50213133630450
