L(s) = 1 | + 1.80·2-s − 3-s + 2.24·4-s − 5-s − 1.80·6-s + 2.24·8-s + 9-s − 1.80·10-s − 2.24·12-s + 15-s + 1.80·16-s + 0.445·17-s + 1.80·18-s + 1.24·19-s − 2.24·20-s + 1.80·23-s − 2.24·24-s + 25-s − 27-s + 1.80·30-s − 0.445·31-s + 1.00·32-s + 0.801·34-s + 2.24·36-s + 2.24·38-s − 2.24·40-s − 45-s + ⋯ |
L(s) = 1 | + 1.80·2-s − 3-s + 2.24·4-s − 5-s − 1.80·6-s + 2.24·8-s + 9-s − 1.80·10-s − 2.24·12-s + 15-s + 1.80·16-s + 0.445·17-s + 1.80·18-s + 1.24·19-s − 2.24·20-s + 1.80·23-s − 2.24·24-s + 25-s − 27-s + 1.80·30-s − 0.445·31-s + 1.00·32-s + 0.801·34-s + 2.24·36-s + 2.24·38-s − 2.24·40-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.370898723\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.370898723\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 - 0.445T + T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - 1.80T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 0.445T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.24T + T^{2} \) |
| 53 | \( 1 + 1.24T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 0.445T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 - 0.445T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.149024019894126891310547610586, −7.81265949916413100940214413217, −7.19203191955541882612169552318, −6.66137847634595940825206819648, −5.68606098599878600761922642637, −5.07812784203999207006784046182, −4.51204682782696918861249758021, −3.59178786686188599465082828773, −2.94606221772665550726257358419, −1.30866428557502754584162577446,
1.30866428557502754584162577446, 2.94606221772665550726257358419, 3.59178786686188599465082828773, 4.51204682782696918861249758021, 5.07812784203999207006784046182, 5.68606098599878600761922642637, 6.66137847634595940825206819648, 7.19203191955541882612169552318, 7.81265949916413100940214413217, 9.149024019894126891310547610586