| L(s) = 1 | − 1.88·2-s + 1.55·4-s − 1.70·7-s + 0.843·8-s + 2.85·11-s − 6.22·13-s + 3.22·14-s − 4.69·16-s − 1.40·17-s + 6.74·19-s − 5.37·22-s + 1.42·23-s + 11.7·26-s − 2.65·28-s − 29-s − 1.29·31-s + 7.16·32-s + 2.64·34-s + 0.989·37-s − 12.7·38-s − 6.32·41-s − 6.03·43-s + 4.42·44-s − 2.67·46-s + 8.97·47-s − 4.07·49-s − 9.65·52-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.776·4-s − 0.645·7-s + 0.298·8-s + 0.859·11-s − 1.72·13-s + 0.860·14-s − 1.17·16-s − 0.340·17-s + 1.54·19-s − 1.14·22-s + 0.296·23-s + 2.29·26-s − 0.501·28-s − 0.185·29-s − 0.232·31-s + 1.26·32-s + 0.453·34-s + 0.162·37-s − 2.06·38-s − 0.987·41-s − 0.920·43-s + 0.667·44-s − 0.395·46-s + 1.30·47-s − 0.582·49-s − 1.33·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 31 | \( 1 + 1.29T + 31T^{2} \) |
| 37 | \( 1 - 0.989T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 8.97T + 47T^{2} \) |
| 53 | \( 1 - 2.11T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 5.73T + 61T^{2} \) |
| 67 | \( 1 - 2.45T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 + 5.55T + 73T^{2} \) |
| 79 | \( 1 - 9.37T + 79T^{2} \) |
| 83 | \( 1 - 5.06T + 83T^{2} \) |
| 89 | \( 1 - 1.75T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.71156608761787966663473671349, −6.95773189249016534951860335205, −6.81653080654946286041306408276, −5.54042382728503136804026488468, −4.86604047966318953919667409167, −3.92913199049652372904232089117, −2.95954807655101570700128708896, −2.07063000141996437858675998767, −1.03860224928909594135275134159, 0,
1.03860224928909594135275134159, 2.07063000141996437858675998767, 2.95954807655101570700128708896, 3.92913199049652372904232089117, 4.86604047966318953919667409167, 5.54042382728503136804026488468, 6.81653080654946286041306408276, 6.95773189249016534951860335205, 7.71156608761787966663473671349
