L(s) = 1 | − 2.14·2-s + 2.58·4-s + 2.79·5-s − 2.93·7-s − 1.25·8-s − 5.99·10-s − 1.15·11-s + 4.41·13-s + 6.27·14-s − 2.48·16-s + 6.41·19-s + 7.23·20-s + 2.48·22-s − 2.79·23-s + 2.82·25-s − 9.45·26-s − 7.57·28-s − 9.55·29-s + 7.83·31-s + 7.83·32-s − 8.19·35-s − 1.39·37-s − 13.7·38-s − 3.50·40-s + 0.480·41-s + 8.07·43-s − 2.99·44-s + ⋯ |
L(s) = 1 | − 1.51·2-s + 1.29·4-s + 1.25·5-s − 1.10·7-s − 0.443·8-s − 1.89·10-s − 0.349·11-s + 1.22·13-s + 1.67·14-s − 0.621·16-s + 1.47·19-s + 1.61·20-s + 0.529·22-s − 0.583·23-s + 0.565·25-s − 1.85·26-s − 1.43·28-s − 1.77·29-s + 1.40·31-s + 1.38·32-s − 1.38·35-s − 0.230·37-s − 2.22·38-s − 0.554·40-s + 0.0749·41-s + 1.23·43-s − 0.451·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9492649336\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9492649336\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 + 2.79T + 23T^{2} \) |
| 29 | \( 1 + 9.55T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 - 0.480T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 - 6.05T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.42T + 59T^{2} \) |
| 61 | \( 1 - 0.765T + 61T^{2} \) |
| 67 | \( 1 - 0.585T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
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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
−9.199447490067227938996482103656, −8.248220923923704536340301758283, −7.54341847450070772339922513962, −6.63899554714074536325482729866, −6.05688280835954651144371407434, −5.33359311573111340995107102276, −3.83064456385572029254802696730, −2.78192789151636231912830142056, −1.81500707767230118301835283509, −0.78751591495338882540573691967,
0.78751591495338882540573691967, 1.81500707767230118301835283509, 2.78192789151636231912830142056, 3.83064456385572029254802696730, 5.33359311573111340995107102276, 6.05688280835954651144371407434, 6.63899554714074536325482729866, 7.54341847450070772339922513962, 8.248220923923704536340301758283, 9.199447490067227938996482103656