L(s) = 1 | − 1.13·2-s − 0.714·4-s + 0.407·7-s + 3.07·8-s − 2·11-s − 0.700·13-s − 0.461·14-s − 2.05·16-s − 1.58·17-s + 4.95·19-s + 2.26·22-s − 1.20·23-s + 0.794·26-s − 0.291·28-s − 5.50·29-s + 8.20·31-s − 3.82·32-s + 1.79·34-s − 5.13·37-s − 5.61·38-s − 7.21·41-s + 9.16·43-s + 1.42·44-s + 1.36·46-s + 1.27·47-s − 6.83·49-s + 0.500·52-s + ⋯ |
L(s) = 1 | − 0.801·2-s − 0.357·4-s + 0.153·7-s + 1.08·8-s − 0.603·11-s − 0.194·13-s − 0.123·14-s − 0.514·16-s − 0.383·17-s + 1.13·19-s + 0.483·22-s − 0.250·23-s + 0.155·26-s − 0.0549·28-s − 1.02·29-s + 1.47·31-s − 0.675·32-s + 0.307·34-s − 0.844·37-s − 0.910·38-s − 1.12·41-s + 1.39·43-s + 0.215·44-s + 0.200·46-s + 0.185·47-s − 0.976·49-s + 0.0694·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 1.13T + 2T^{2} \) |
| 7 | \( 1 - 0.407T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 0.700T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 8.20T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 - 9.16T + 43T^{2} \) |
| 47 | \( 1 - 1.27T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 - 6.49T + 59T^{2} \) |
| 61 | \( 1 + 9.42T + 61T^{2} \) |
| 67 | \( 1 + 3.08T + 67T^{2} \) |
| 71 | \( 1 + 6.86T + 71T^{2} \) |
| 73 | \( 1 - 0.545T + 73T^{2} \) |
| 79 | \( 1 - 5.48T + 79T^{2} \) |
| 83 | \( 1 + 0.974T + 83T^{2} \) |
| 89 | \( 1 - 2.26T + 89T^{2} \) |
| 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
−7.74868311811428539154004616313, −7.44623074046196390436902119994, −6.50178609185634199507794911519, −5.49840493974893463816167177903, −4.93000047699181294802374460701, −4.14903910446306345102200866298, −3.19674318074098773502950715540, −2.14666020963079229262910118541, −1.13450562544769550004954571627, 0,
1.13450562544769550004954571627, 2.14666020963079229262910118541, 3.19674318074098773502950715540, 4.14903910446306345102200866298, 4.93000047699181294802374460701, 5.49840493974893463816167177903, 6.50178609185634199507794911519, 7.44623074046196390436902119994, 7.74868311811428539154004616313