| L(s) = 1 | + 2-s + 4-s − 2.56·5-s − 7-s + 8-s − 2.56·10-s + 3.12·11-s + 0.561·13-s − 14-s + 16-s − 7.12·17-s + 3.12·19-s − 2.56·20-s + 3.12·22-s − 23-s + 1.56·25-s + 0.561·26-s − 28-s − 3.43·29-s − 5.12·31-s + 32-s − 7.12·34-s + 2.56·35-s + 2.56·37-s + 3.12·38-s − 2.56·40-s + 0.561·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.14·5-s − 0.377·7-s + 0.353·8-s − 0.810·10-s + 0.941·11-s + 0.155·13-s − 0.267·14-s + 0.250·16-s − 1.72·17-s + 0.716·19-s − 0.572·20-s + 0.665·22-s − 0.208·23-s + 0.312·25-s + 0.110·26-s − 0.188·28-s − 0.638·29-s − 0.920·31-s + 0.176·32-s − 1.22·34-s + 0.432·35-s + 0.421·37-s + 0.506·38-s − 0.405·40-s + 0.0876·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.56T + 5T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 0.876T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.328872305181468231792344693402, −7.45061820328268051734395708104, −6.84437884226441358308352963666, −6.18020019111576913208434725983, −5.16837199052561964432297922465, −4.19864030745313188478856111238, −3.82376625783524314410067799514, −2.88593429190987833786608650985, −1.63328528299370640231478682293, 0,
1.63328528299370640231478682293, 2.88593429190987833786608650985, 3.82376625783524314410067799514, 4.19864030745313188478856111238, 5.16837199052561964432297922465, 6.18020019111576913208434725983, 6.84437884226441358308352963666, 7.45061820328268051734395708104, 8.328872305181468231792344693402
