L(s) = 1 | − 2.23·2-s + 3-s + 3.00·4-s − 2.23·6-s − 0.583·7-s − 2.25·8-s + 9-s − 2.62·11-s + 3.00·12-s + 0.671·13-s + 1.30·14-s − 0.965·16-s + 6.30·17-s − 2.23·18-s + 1.49·19-s − 0.583·21-s + 5.88·22-s + 5.58·23-s − 2.25·24-s − 1.50·26-s + 27-s − 1.75·28-s + 0.925·29-s − 7.27·31-s + 6.67·32-s − 2.62·33-s − 14.1·34-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.50·4-s − 0.913·6-s − 0.220·7-s − 0.798·8-s + 0.333·9-s − 0.792·11-s + 0.868·12-s + 0.186·13-s + 0.349·14-s − 0.241·16-s + 1.52·17-s − 0.527·18-s + 0.344·19-s − 0.127·21-s + 1.25·22-s + 1.16·23-s − 0.460·24-s − 0.294·26-s + 0.192·27-s − 0.332·28-s + 0.171·29-s − 1.30·31-s + 1.18·32-s − 0.457·33-s − 2.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.006001653\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.006001653\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 + 0.583T + 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 - 0.671T + 13T^{2} \) |
| 17 | \( 1 - 6.30T + 17T^{2} \) |
| 19 | \( 1 - 1.49T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 0.925T + 29T^{2} \) |
| 31 | \( 1 + 7.27T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 53 | \( 1 + 4.64T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 5.77T + 73T^{2} \) |
| 79 | \( 1 + 9.02T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 - 6.50T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.601695490407767304576335633767, −7.898035603875292523674961396253, −7.48075624279938916415177575221, −6.76671895903300704259291275353, −5.71405197048224929478110540437, −4.84459269105202005360429120854, −3.51665378107879998218391921555, −2.80071134551960250692214884532, −1.74330917516982720156619154007, −0.74814333354838498264641361224,
0.74814333354838498264641361224, 1.74330917516982720156619154007, 2.80071134551960250692214884532, 3.51665378107879998218391921555, 4.84459269105202005360429120854, 5.71405197048224929478110540437, 6.76671895903300704259291275353, 7.48075624279938916415177575221, 7.898035603875292523674961396253, 8.601695490407767304576335633767