L(s) = 1 | + 0.676·2-s + 1.15·3-s − 1.54·4-s + 5-s + 0.779·6-s − 4.22·7-s − 2.39·8-s − 1.66·9-s + 0.676·10-s + 11-s − 1.77·12-s − 2.85·14-s + 1.15·15-s + 1.46·16-s − 5.49·17-s − 1.12·18-s − 3.78·19-s − 1.54·20-s − 4.87·21-s + 0.676·22-s + 0.422·23-s − 2.76·24-s + 25-s − 5.38·27-s + 6.52·28-s − 5.97·29-s + 0.779·30-s + ⋯ |
L(s) = 1 | + 0.478·2-s + 0.665·3-s − 0.771·4-s + 0.447·5-s + 0.318·6-s − 1.59·7-s − 0.846·8-s − 0.556·9-s + 0.213·10-s + 0.301·11-s − 0.513·12-s − 0.764·14-s + 0.297·15-s + 0.366·16-s − 1.33·17-s − 0.266·18-s − 0.868·19-s − 0.345·20-s − 1.06·21-s + 0.144·22-s + 0.0882·23-s − 0.563·24-s + 0.200·25-s − 1.03·27-s + 1.23·28-s − 1.10·29-s + 0.142·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.051137056\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.051137056\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.676T + 2T^{2} \) |
| 3 | \( 1 - 1.15T + 3T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 23 | \( 1 - 0.422T + 23T^{2} \) |
| 29 | \( 1 + 5.97T + 29T^{2} \) |
| 31 | \( 1 - 0.372T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 - 5.96T + 41T^{2} \) |
| 43 | \( 1 - 9.82T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 2.31T + 53T^{2} \) |
| 59 | \( 1 - 4.65T + 59T^{2} \) |
| 61 | \( 1 + 3.08T + 61T^{2} \) |
| 67 | \( 1 - 0.431T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 - 1.36T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.76856516610466790321253022711, −6.82796517925317585012625440922, −6.20551710480727347039870709733, −5.80970423664310549990649888355, −4.87026398648254668610599914767, −4.01488302152492166588203752766, −3.51989323289799536512701823222, −2.77525686310637817949099194543, −2.08118811988922462137648537719, −0.41320284562080967313338061815,
0.41320284562080967313338061815, 2.08118811988922462137648537719, 2.77525686310637817949099194543, 3.51989323289799536512701823222, 4.01488302152492166588203752766, 4.87026398648254668610599914767, 5.80970423664310549990649888355, 6.20551710480727347039870709733, 6.82796517925317585012625440922, 7.76856516610466790321253022711