| L(s) = 1 | + 1.08·2-s − 0.812·4-s + 3.08·7-s − 3.06·8-s + 1.14·11-s − 4.07·13-s + 3.36·14-s − 1.71·16-s + 4.62·17-s − 5.96·19-s + 1.25·22-s − 2.32·23-s − 4.44·26-s − 2.50·28-s + 5.28·29-s − 0.589·31-s + 4.26·32-s + 5.04·34-s − 11.3·37-s − 6.50·38-s − 9.49·41-s − 2.42·43-s − 0.932·44-s − 2.53·46-s + 6.04·47-s + 2.53·49-s + 3.31·52-s + ⋯ |
| L(s) = 1 | + 0.770·2-s − 0.406·4-s + 1.16·7-s − 1.08·8-s + 0.346·11-s − 1.13·13-s + 0.899·14-s − 0.428·16-s + 1.12·17-s − 1.36·19-s + 0.266·22-s − 0.484·23-s − 0.871·26-s − 0.473·28-s + 0.981·29-s − 0.105·31-s + 0.753·32-s + 0.864·34-s − 1.87·37-s − 1.05·38-s − 1.48·41-s − 0.370·43-s − 0.140·44-s − 0.373·46-s + 0.882·47-s + 0.361·49-s + 0.459·52-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 | \( 1 - 1.08T + 2T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 + 5.96T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 31 | \( 1 + 0.589T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 + 2.42T + 43T^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 + 3.18T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 + 6.46T + 71T^{2} \) |
| 73 | \( 1 + 7.20T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 1.08T + 89T^{2} \) |
| 97 | \( 1 + 4.52T + 97T^{2} \) |
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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
−7.84131924773635554721058079796, −6.99446799615257046261876676769, −6.21063044758948984575277394556, −5.30016797340848663243584511136, −4.92862631689349191734342909609, −4.21887923106639408961242889605, −3.46196269590455854044740922474, −2.45582049848557124046809373933, −1.48154711135775154691756126099, 0,
1.48154711135775154691756126099, 2.45582049848557124046809373933, 3.46196269590455854044740922474, 4.21887923106639408961242889605, 4.92862631689349191734342909609, 5.30016797340848663243584511136, 6.21063044758948984575277394556, 6.99446799615257046261876676769, 7.84131924773635554721058079796
