| L(s) = 1 | + 1.79·2-s + 2.17·3-s + 1.21·4-s − 4.05·5-s + 3.90·6-s + 2.54·7-s − 1.40·8-s + 1.74·9-s − 7.26·10-s − 4.74·11-s + 2.64·12-s + 4.55·14-s − 8.82·15-s − 4.95·16-s + 2.22·17-s + 3.11·18-s + 19-s − 4.92·20-s + 5.53·21-s − 8.51·22-s − 1.82·23-s − 3.06·24-s + 11.4·25-s − 2.74·27-s + 3.08·28-s − 0.466·29-s − 15.8·30-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 1.25·3-s + 0.607·4-s − 1.81·5-s + 1.59·6-s + 0.960·7-s − 0.497·8-s + 0.580·9-s − 2.29·10-s − 1.43·11-s + 0.763·12-s + 1.21·14-s − 2.27·15-s − 1.23·16-s + 0.538·17-s + 0.735·18-s + 0.229·19-s − 1.10·20-s + 1.20·21-s − 1.81·22-s − 0.380·23-s − 0.625·24-s + 2.28·25-s − 0.527·27-s + 0.583·28-s − 0.0866·29-s − 2.88·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 + 4.05T + 5T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 + 0.466T + 29T^{2} \) |
| 31 | \( 1 + 9.97T + 31T^{2} \) |
| 37 | \( 1 + 3.79T + 37T^{2} \) |
| 41 | \( 1 - 3.99T + 41T^{2} \) |
| 43 | \( 1 + 0.0540T + 43T^{2} \) |
| 47 | \( 1 + 8.09T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 - 1.16T + 61T^{2} \) |
| 67 | \( 1 + 6.78T + 67T^{2} \) |
| 71 | \( 1 + 5.56T + 71T^{2} \) |
| 73 | \( 1 - 6.14T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 8.24T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 6.45T + 97T^{2} \) |
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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
−8.001710922511923879946896732350, −7.81360684841984569037723555091, −7.03833146968221336170722013483, −5.64736110259111847502045335967, −4.92727257600763131877031596892, −4.31264907118411733504715935490, −3.41089014749524503252229164622, −3.14545762754968710771695997844, −2.00625788462846071320525826017, 0,
2.00625788462846071320525826017, 3.14545762754968710771695997844, 3.41089014749524503252229164622, 4.31264907118411733504715935490, 4.92727257600763131877031596892, 5.64736110259111847502045335967, 7.03833146968221336170722013483, 7.81360684841984569037723555091, 8.001710922511923879946896732350
