L(s) = 1 | + 2.96·3-s + 1.23·5-s + 4.80·7-s + 5.80·9-s − 4.80·11-s + 3.66·15-s − 17-s + 5.13·19-s + 14.2·21-s − 3.96·23-s − 3.47·25-s + 8.32·27-s + 1.33·29-s + 3.66·31-s − 14.2·33-s + 5.93·35-s − 3.62·37-s + 7.66·41-s − 2.38·43-s + 7.16·45-s + 5.94·47-s + 16.0·49-s − 2.96·51-s + 0.139·53-s − 5.93·55-s + 15.2·57-s + 5.47·59-s + ⋯ |
L(s) = 1 | + 1.71·3-s + 0.552·5-s + 1.81·7-s + 1.93·9-s − 1.44·11-s + 0.946·15-s − 0.242·17-s + 1.17·19-s + 3.11·21-s − 0.825·23-s − 0.694·25-s + 1.60·27-s + 0.247·29-s + 0.658·31-s − 2.48·33-s + 1.00·35-s − 0.595·37-s + 1.19·41-s − 0.364·43-s + 1.06·45-s + 0.867·47-s + 2.29·49-s − 0.415·51-s + 0.0191·53-s − 0.800·55-s + 2.01·57-s + 0.712·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.137554272\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.137554272\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 4.80T + 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 5.13T + 19T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 - 1.33T + 29T^{2} \) |
| 31 | \( 1 - 3.66T + 31T^{2} \) |
| 37 | \( 1 + 3.62T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 - 5.94T + 47T^{2} \) |
| 53 | \( 1 - 0.139T + 53T^{2} \) |
| 59 | \( 1 - 5.47T + 59T^{2} \) |
| 61 | \( 1 - 4.94T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 + 1.13T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 + 4.77T + 79T^{2} \) |
| 83 | \( 1 + 1.60T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.102098829400259032741422226381, −7.73848525305371960471728038311, −7.21601879929890544657723196023, −5.85574625251574074579957445111, −5.13746139800441031827460045007, −4.46892552812076689336983611688, −3.59264457097128764747637843461, −2.50739878676306923487312739145, −2.19134061192460131251088737841, −1.24631448189815198334064489324,
1.24631448189815198334064489324, 2.19134061192460131251088737841, 2.50739878676306923487312739145, 3.59264457097128764747637843461, 4.46892552812076689336983611688, 5.13746139800441031827460045007, 5.85574625251574074579957445111, 7.21601879929890544657723196023, 7.73848525305371960471728038311, 8.102098829400259032741422226381