L(s) = 1 | − 2-s + 2.10·3-s + 4-s + 4.16·5-s − 2.10·6-s − 7-s − 8-s + 1.41·9-s − 4.16·10-s − 3.72·11-s + 2.10·12-s − 3.37·13-s + 14-s + 8.75·15-s + 16-s + 1.77·17-s − 1.41·18-s + 0.724·19-s + 4.16·20-s − 2.10·21-s + 3.72·22-s + 1.41·23-s − 2.10·24-s + 12.3·25-s + 3.37·26-s − 3.33·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.21·3-s + 0.5·4-s + 1.86·5-s − 0.857·6-s − 0.377·7-s − 0.353·8-s + 0.471·9-s − 1.31·10-s − 1.12·11-s + 0.606·12-s − 0.934·13-s + 0.267·14-s + 2.26·15-s + 0.250·16-s + 0.430·17-s − 0.333·18-s + 0.166·19-s + 0.931·20-s − 0.458·21-s + 0.795·22-s + 0.294·23-s − 0.428·24-s + 2.47·25-s + 0.661·26-s − 0.641·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.926120539\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.926120539\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 - 1.77T + 17T^{2} \) |
| 19 | \( 1 - 0.724T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 0.0884T + 29T^{2} \) |
| 31 | \( 1 - 6.94T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 - 8.67T + 41T^{2} \) |
| 43 | \( 1 - 5.90T + 43T^{2} \) |
| 47 | \( 1 - 9.49T + 47T^{2} \) |
| 53 | \( 1 + 0.0494T + 53T^{2} \) |
| 59 | \( 1 - 9.33T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 - 4.62T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 4.76T + 79T^{2} \) |
| 83 | \( 1 + 5.90T + 83T^{2} \) |
| 89 | \( 1 + 0.395T + 89T^{2} \) |
| 97 | \( 1 - 2.82T + 97T^{2} \) |
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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
−8.227607768501390005884183378772, −7.47773846080084615947007280951, −6.89266747366746721896157330316, −5.83300331863368148764939989831, −5.54641152638462950985556331353, −4.44625275851105727920911241196, −3.04369505760300258845432342925, −2.56261419375124475929545890678, −2.17224628559125214106977477244, −0.938636392979836618123654551160,
0.938636392979836618123654551160, 2.17224628559125214106977477244, 2.56261419375124475929545890678, 3.04369505760300258845432342925, 4.44625275851105727920911241196, 5.54641152638462950985556331353, 5.83300331863368148764939989831, 6.89266747366746721896157330316, 7.47773846080084615947007280951, 8.227607768501390005884183378772