| L(s) = 1 | + 1.76·2-s + 3-s + 1.10·4-s − 3.70·5-s + 1.76·6-s − 1.57·8-s + 9-s − 6.53·10-s − 1.53·11-s + 1.10·12-s + 2.45·13-s − 3.70·15-s − 4.98·16-s + 4.82·17-s + 1.76·18-s + 3.81·19-s − 4.11·20-s − 2.70·22-s + 4.81·23-s − 1.57·24-s + 8.75·25-s + 4.32·26-s + 27-s − 0.324·29-s − 6.53·30-s − 6.64·31-s − 5.65·32-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.577·3-s + 0.554·4-s − 1.65·5-s + 0.719·6-s − 0.555·8-s + 0.333·9-s − 2.06·10-s − 0.462·11-s + 0.320·12-s + 0.680·13-s − 0.957·15-s − 1.24·16-s + 1.17·17-s + 0.415·18-s + 0.874·19-s − 0.919·20-s − 0.576·22-s + 1.00·23-s − 0.320·24-s + 1.75·25-s + 0.847·26-s + 0.192·27-s − 0.0602·29-s − 1.19·30-s − 1.19·31-s − 0.999·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.76T + 2T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 2.45T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 3.81T + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 + 0.324T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 + 5.44T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 0.829T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.48091856212963561264887505659, −7.03221580574228165105442764856, −5.99045851092036808714532378090, −5.18968972953981197071122735672, −4.66873525388741942138675598326, −3.77027653913391913921821154420, −3.36060113551920986628724795001, −2.93257298089630375188565483995, −1.39751994061843374540059342014, 0,
1.39751994061843374540059342014, 2.93257298089630375188565483995, 3.36060113551920986628724795001, 3.77027653913391913921821154420, 4.66873525388741942138675598326, 5.18968972953981197071122735672, 5.99045851092036808714532378090, 7.03221580574228165105442764856, 7.48091856212963561264887505659
