| L(s) = 1 | + 2-s + 4-s + 4.24·5-s + 8-s + 4.24·10-s + 11-s + 5.65·13-s + 16-s − 7.07·17-s + 1.41·19-s + 4.24·20-s + 22-s − 8·23-s + 12.9·25-s + 5.65·26-s − 8·29-s + 4.24·31-s + 32-s − 7.07·34-s + 2·37-s + 1.41·38-s + 4.24·40-s + 1.41·41-s + 8·43-s + 44-s − 8·46-s + 9.89·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.89·5-s + 0.353·8-s + 1.34·10-s + 0.301·11-s + 1.56·13-s + 0.250·16-s − 1.71·17-s + 0.324·19-s + 0.948·20-s + 0.213·22-s − 1.66·23-s + 2.59·25-s + 1.10·26-s − 1.48·29-s + 0.762·31-s + 0.176·32-s − 1.21·34-s + 0.328·37-s + 0.229·38-s + 0.670·40-s + 0.220·41-s + 1.21·43-s + 0.150·44-s − 1.17·46-s + 1.44·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.546588717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.546588717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 4.24T + 5T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.52204659541822019350827276698, −6.57182712994077345858664608011, −6.12632269854921076955983445249, −5.88784218459954431809309619784, −5.01402616166623662589998768862, −4.19992003205051542010758706180, −3.52839724529404755240780040537, −2.32917435624859505691503200915, −2.04793533343927042872727784676, −1.05982390916483838834168913622,
1.05982390916483838834168913622, 2.04793533343927042872727784676, 2.32917435624859505691503200915, 3.52839724529404755240780040537, 4.19992003205051542010758706180, 5.01402616166623662589998768862, 5.88784218459954431809309619784, 6.12632269854921076955983445249, 6.57182712994077345858664608011, 7.52204659541822019350827276698
