L(s) = 1 | + 2.49·3-s + 0.494·5-s + 2.18·7-s + 3.23·9-s + 3.30·11-s + 3.85·13-s + 1.23·15-s − 2.84·17-s − 0.107·19-s + 5.46·21-s − 0.225·23-s − 4.75·25-s + 0.582·27-s − 7.21·29-s − 1.87·31-s + 8.24·33-s + 1.08·35-s − 5.20·37-s + 9.63·39-s + 6.59·41-s − 6.49·43-s + 1.59·45-s + 9.83·47-s − 2.20·49-s − 7.11·51-s + 3.68·53-s + 1.63·55-s + ⋯ |
L(s) = 1 | + 1.44·3-s + 0.221·5-s + 0.827·7-s + 1.07·9-s + 0.995·11-s + 1.06·13-s + 0.318·15-s − 0.691·17-s − 0.0245·19-s + 1.19·21-s − 0.0470·23-s − 0.951·25-s + 0.112·27-s − 1.34·29-s − 0.336·31-s + 1.43·33-s + 0.183·35-s − 0.855·37-s + 1.54·39-s + 1.02·41-s − 0.990·43-s + 0.238·45-s + 1.43·47-s − 0.315·49-s − 0.996·51-s + 0.506·53-s + 0.220·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.257755269\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.257755269\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 3 | \( 1 - 2.49T + 3T^{2} \) |
| 5 | \( 1 - 0.494T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 + 0.107T + 19T^{2} \) |
| 23 | \( 1 + 0.225T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 - 9.83T + 47T^{2} \) |
| 53 | \( 1 - 3.68T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 5.59T + 61T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 - 5.33T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 + 6.02T + 79T^{2} \) |
| 83 | \( 1 - 4.72T + 83T^{2} \) |
| 89 | \( 1 - 0.842T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.190567770123026426937136245351, −8.943298349022393508523261063261, −8.129007619860827307639223934677, −7.42752310936290210049396360254, −6.43301620810986064419396025926, −5.42046660929046056045582004602, −4.07907339783622369969991297380, −3.64874401017678139936554284305, −2.28053199297104851177852492938, −1.51788712502748915716106788013,
1.51788712502748915716106788013, 2.28053199297104851177852492938, 3.64874401017678139936554284305, 4.07907339783622369969991297380, 5.42046660929046056045582004602, 6.43301620810986064419396025926, 7.42752310936290210049396360254, 8.129007619860827307639223934677, 8.943298349022393508523261063261, 9.190567770123026426937136245351