| L(s) = 1 | − 2.95·3-s − 0.939·7-s + 5.74·9-s − 1.84·11-s − 6.00·13-s − 5.08·17-s + 3.73·19-s + 2.77·21-s + 4.23·23-s − 8.13·27-s + 0.310·29-s + 5.50·31-s + 5.45·33-s + 37-s + 17.7·39-s + 5.16·41-s + 8.26·43-s − 11.1·47-s − 6.11·49-s + 15.0·51-s + 12.5·53-s − 11.0·57-s − 10.1·59-s − 12.1·61-s − 5.40·63-s + 8.40·67-s − 12.5·69-s + ⋯ |
| L(s) = 1 | − 1.70·3-s − 0.355·7-s + 1.91·9-s − 0.555·11-s − 1.66·13-s − 1.23·17-s + 0.856·19-s + 0.606·21-s + 0.882·23-s − 1.56·27-s + 0.0576·29-s + 0.988·31-s + 0.948·33-s + 0.164·37-s + 2.84·39-s + 0.806·41-s + 1.26·43-s − 1.63·47-s − 0.873·49-s + 2.10·51-s + 1.72·53-s − 1.46·57-s − 1.32·59-s − 1.55·61-s − 0.680·63-s + 1.02·67-s − 1.50·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.95T + 3T^{2} \) |
| 7 | \( 1 + 0.939T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 - 0.310T + 29T^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 41 | \( 1 - 5.16T + 41T^{2} \) |
| 43 | \( 1 - 8.26T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 5.06T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 6.69T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 2.46T + 97T^{2} \) |
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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
−7.32130639204632746425349171854, −6.75876325076643497393531436269, −6.17960602560708942219713889052, −5.36406284341542676887082043714, −4.83910852924631431464483752500, −4.38621686587478098031226099260, −3.06323099618941609541453997654, −2.20968707240768147442259439429, −0.879741498050797199283884864291, 0,
0.879741498050797199283884864291, 2.20968707240768147442259439429, 3.06323099618941609541453997654, 4.38621686587478098031226099260, 4.83910852924631431464483752500, 5.36406284341542676887082043714, 6.17960602560708942219713889052, 6.75876325076643497393531436269, 7.32130639204632746425349171854
