| L(s) = 1 | − 0.0249·2-s − 3-s − 1.99·4-s − 2.90·5-s + 0.0249·6-s + 1.03·7-s + 0.0995·8-s + 9-s + 0.0724·10-s + 4.05·11-s + 1.99·12-s + 6.69·13-s − 0.0258·14-s + 2.90·15-s + 3.99·16-s + 17-s − 0.0249·18-s − 2.37·19-s + 5.81·20-s − 1.03·21-s − 0.101·22-s − 2.49·23-s − 0.0995·24-s + 3.46·25-s − 0.166·26-s − 27-s − 2.07·28-s + ⋯ |
| L(s) = 1 | − 0.0176·2-s − 0.577·3-s − 0.999·4-s − 1.30·5-s + 0.0101·6-s + 0.391·7-s + 0.0352·8-s + 0.333·9-s + 0.0229·10-s + 1.22·11-s + 0.577·12-s + 1.85·13-s − 0.00689·14-s + 0.751·15-s + 0.999·16-s + 0.242·17-s − 0.00586·18-s − 0.544·19-s + 1.30·20-s − 0.226·21-s − 0.0215·22-s − 0.520·23-s − 0.0203·24-s + 0.693·25-s − 0.0327·26-s − 0.192·27-s − 0.391·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9711073059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9711073059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0249T + 2T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 - 4.05T + 11T^{2} \) |
| 13 | \( 1 - 6.69T + 13T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 + 7.67T + 29T^{2} \) |
| 31 | \( 1 + 0.454T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 5.84T + 41T^{2} \) |
| 43 | \( 1 + 1.58T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 - 1.80T + 53T^{2} \) |
| 59 | \( 1 + 2.48T + 59T^{2} \) |
| 61 | \( 1 - 1.40T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 7.25T + 73T^{2} \) |
| 83 | \( 1 + 9.84T + 83T^{2} \) |
| 89 | \( 1 + 0.599T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.230576506584355062166775625996, −8.048743251805191099842048457404, −6.96480227038021658651801292530, −6.15923167199954807211233561921, −5.47768376961663221019056304716, −4.39559677019473208000154451375, −3.95573479148853948651873129045, −3.50720851512645325734511819045, −1.55661855469455673653929690864, −0.63867534930002280089578292725,
0.63867534930002280089578292725, 1.55661855469455673653929690864, 3.50720851512645325734511819045, 3.95573479148853948651873129045, 4.39559677019473208000154451375, 5.47768376961663221019056304716, 6.15923167199954807211233561921, 6.96480227038021658651801292530, 8.048743251805191099842048457404, 8.230576506584355062166775625996
