| L(s) = 1 | + 0.347·2-s − 3-s − 1.87·4-s − 1.45·5-s − 0.347·6-s − 2.71·7-s − 1.34·8-s + 9-s − 0.506·10-s + 1.87·12-s + 1.83·13-s − 0.944·14-s + 1.45·15-s + 3.28·16-s − 17-s + 0.347·18-s − 5.64·19-s + 2.74·20-s + 2.71·21-s + 2.86·23-s + 1.34·24-s − 2.87·25-s + 0.636·26-s − 27-s + 5.10·28-s − 0.687·29-s + 0.506·30-s + ⋯ |
| L(s) = 1 | + 0.245·2-s − 0.577·3-s − 0.939·4-s − 0.652·5-s − 0.141·6-s − 1.02·7-s − 0.476·8-s + 0.333·9-s − 0.160·10-s + 0.542·12-s + 0.508·13-s − 0.252·14-s + 0.376·15-s + 0.822·16-s − 0.242·17-s + 0.0819·18-s − 1.29·19-s + 0.612·20-s + 0.592·21-s + 0.597·23-s + 0.275·24-s − 0.574·25-s + 0.124·26-s − 0.192·27-s + 0.964·28-s − 0.127·29-s + 0.0925·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 0.347T + 2T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 + 0.687T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 - 9.38T + 37T^{2} \) |
| 41 | \( 1 - 0.342T + 41T^{2} \) |
| 43 | \( 1 - 0.261T + 43T^{2} \) |
| 47 | \( 1 - 3.53T + 47T^{2} \) |
| 53 | \( 1 + 7.22T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 + 0.296T + 71T^{2} \) |
| 73 | \( 1 - 0.0393T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 - 7.54T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 0.645T + 97T^{2} \) |
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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
−7.77616849354285094926980798730, −6.77981438827713594164314512486, −6.25556479446354699925334274771, −5.59684865424052539175401408280, −4.67885773871818093021079361673, −4.06931666519335506289434616596, −3.52797086098515660450031741680, −2.48657754053066598105568525801, −0.901388798758029617951927282359, 0,
0.901388798758029617951927282359, 2.48657754053066598105568525801, 3.52797086098515660450031741680, 4.06931666519335506289434616596, 4.67885773871818093021079361673, 5.59684865424052539175401408280, 6.25556479446354699925334274771, 6.77981438827713594164314512486, 7.77616849354285094926980798730
