L(s) = 1 | − 0.0950·3-s − 3.14·7-s − 2.99·9-s − 1.67·11-s − 6.63·13-s − 5.16·17-s − 4.72·19-s + 0.299·21-s + 8.82·23-s + 0.569·27-s − 1.80·29-s − 1.65·31-s + 0.159·33-s + 1.99·37-s + 0.630·39-s − 41-s − 1.46·43-s + 8.53·47-s + 2.89·49-s + 0.490·51-s + 9.35·53-s + 0.449·57-s + 8.82·59-s + 12.6·61-s + 9.40·63-s − 9.67·67-s − 0.838·69-s + ⋯ |
L(s) = 1 | − 0.0549·3-s − 1.18·7-s − 0.996·9-s − 0.505·11-s − 1.83·13-s − 1.25·17-s − 1.08·19-s + 0.0652·21-s + 1.83·23-s + 0.109·27-s − 0.336·29-s − 0.298·31-s + 0.0277·33-s + 0.327·37-s + 0.100·39-s − 0.156·41-s − 0.224·43-s + 1.24·47-s + 0.413·49-s + 0.0687·51-s + 1.28·53-s + 0.0595·57-s + 1.14·59-s + 1.61·61-s + 1.18·63-s − 1.18·67-s − 0.100·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5444466497\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5444466497\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 0.0950T + 3T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 6.63T + 13T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 - 1.99T + 37T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 - 9.35T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 9.67T + 67T^{2} \) |
| 71 | \( 1 + 0.776T + 71T^{2} \) |
| 73 | \( 1 + 8.33T + 73T^{2} \) |
| 79 | \( 1 - 0.915T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 - 8.32T + 97T^{2} \) |
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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
−8.726112554190236985013457870391, −7.51423186969251079601407793357, −6.96362313383628902816289253849, −6.31410437999165802581352893454, −5.40401607314196164949701511131, −4.79503706498838232178098657904, −3.77734694875441173058225376119, −2.64940511228198175794876545172, −2.43428789886479336996782150842, −0.38491427194733904473738529633,
0.38491427194733904473738529633, 2.43428789886479336996782150842, 2.64940511228198175794876545172, 3.77734694875441173058225376119, 4.79503706498838232178098657904, 5.40401607314196164949701511131, 6.31410437999165802581352893454, 6.96362313383628902816289253849, 7.51423186969251079601407793357, 8.726112554190236985013457870391