| L(s) = 1 | − 16·2-s + 173.·3-s + 256·4-s − 2.34e3·5-s − 2.77e3·6-s − 4.09e3·8-s + 1.04e4·9-s + 3.75e4·10-s − 7.11e4·11-s + 4.44e4·12-s − 8.94e4·13-s − 4.06e5·15-s + 6.55e4·16-s − 3.22e4·17-s − 1.66e5·18-s + 7.79e5·19-s − 6.00e5·20-s + 1.13e6·22-s + 1.77e6·23-s − 7.10e5·24-s + 3.54e6·25-s + 1.43e6·26-s − 1.60e6·27-s − 2.12e5·29-s + 6.50e6·30-s + 4.31e6·31-s − 1.04e6·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.23·3-s + 0.5·4-s − 1.67·5-s − 0.874·6-s − 0.353·8-s + 0.528·9-s + 1.18·10-s − 1.46·11-s + 0.618·12-s − 0.868·13-s − 2.07·15-s + 0.250·16-s − 0.0937·17-s − 0.373·18-s + 1.37·19-s − 0.839·20-s + 1.03·22-s + 1.32·23-s − 0.437·24-s + 1.81·25-s + 0.614·26-s − 0.582·27-s − 0.0556·29-s + 1.46·30-s + 0.838·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.161384436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.161384436\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 173.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.34e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 7.11e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 8.94e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.22e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.79e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.77e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.12e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.31e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.20e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.40e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.99e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.50e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.26e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.90e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.18e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.03e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.91e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.08e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.16e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.86e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.53e8T + 7.60e17T^{2} \) |
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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
−11.97232396460856467950329603073, −10.98290397400760559815581777265, −9.720980271288265715885301673394, −8.585333545435920061567198382296, −7.77441857584402058065383058158, −7.30047239721179268108364973742, −4.93802350007973741938292620359, −3.35618122319118413760806685863, −2.59525381973879630477483977317, −0.60704637627975170907392082485,
0.60704637627975170907392082485, 2.59525381973879630477483977317, 3.35618122319118413760806685863, 4.93802350007973741938292620359, 7.30047239721179268108364973742, 7.77441857584402058065383058158, 8.585333545435920061567198382296, 9.720980271288265715885301673394, 10.98290397400760559815581777265, 11.97232396460856467950329603073
