| L(s) = 1 | + 0.365·3-s + 2.99·5-s + 7-s − 2.86·9-s − 0.171·11-s − 3.29·13-s + 1.09·15-s + 2.16·17-s + 19-s + 0.365·21-s − 5.12·23-s + 3.96·25-s − 2.14·27-s − 9.28·29-s − 0.165·31-s − 0.0628·33-s + 2.99·35-s + 10.4·37-s − 1.20·39-s − 10.9·41-s + 5.86·43-s − 8.58·45-s − 10.7·47-s + 49-s + 0.791·51-s + 0.616·53-s − 0.514·55-s + ⋯ |
| L(s) = 1 | + 0.211·3-s + 1.33·5-s + 0.377·7-s − 0.955·9-s − 0.0518·11-s − 0.913·13-s + 0.282·15-s + 0.525·17-s + 0.229·19-s + 0.0797·21-s − 1.06·23-s + 0.792·25-s − 0.412·27-s − 1.72·29-s − 0.0296·31-s − 0.0109·33-s + 0.505·35-s + 1.72·37-s − 0.192·39-s − 1.70·41-s + 0.894·43-s − 1.27·45-s − 1.56·47-s + 0.142·49-s + 0.110·51-s + 0.0846·53-s − 0.0693·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.365T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 11 | \( 1 + 0.171T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 - 2.16T + 17T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 + 9.28T + 29T^{2} \) |
| 31 | \( 1 + 0.165T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 0.616T + 53T^{2} \) |
| 59 | \( 1 + 8.37T + 59T^{2} \) |
| 61 | \( 1 + 1.85T + 61T^{2} \) |
| 67 | \( 1 + 9.05T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 6.43T + 73T^{2} \) |
| 79 | \( 1 - 5.34T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + 5.83T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.71136016680668657134549670190, −6.59303116539160970002143019177, −5.95760172776423124843515293206, −5.44828069204933159924613994653, −4.85869495405754928195940846555, −3.79550506872092839078183746512, −2.87868977541057684349809946521, −2.20229978909314086126770632356, −1.51701029089982619134666369698, 0,
1.51701029089982619134666369698, 2.20229978909314086126770632356, 2.87868977541057684349809946521, 3.79550506872092839078183746512, 4.85869495405754928195940846555, 5.44828069204933159924613994653, 5.95760172776423124843515293206, 6.59303116539160970002143019177, 7.71136016680668657134549670190
