L(s) = 1 | − 1.73·2-s + 3-s + 0.997·4-s + 5-s − 1.73·6-s + 0.816·7-s + 1.73·8-s + 9-s − 1.73·10-s + 4.42·11-s + 0.997·12-s − 13-s − 1.41·14-s + 15-s − 4.99·16-s − 3.98·17-s − 1.73·18-s − 0.455·19-s + 0.997·20-s + 0.816·21-s − 7.65·22-s − 5.69·23-s + 1.73·24-s + 25-s + 1.73·26-s + 27-s + 0.814·28-s + ⋯ |
L(s) = 1 | − 1.22·2-s + 0.577·3-s + 0.498·4-s + 0.447·5-s − 0.706·6-s + 0.308·7-s + 0.613·8-s + 0.333·9-s − 0.547·10-s + 1.33·11-s + 0.287·12-s − 0.277·13-s − 0.377·14-s + 0.258·15-s − 1.24·16-s − 0.965·17-s − 0.408·18-s − 0.104·19-s + 0.223·20-s + 0.178·21-s − 1.63·22-s − 1.18·23-s + 0.354·24-s + 0.200·25-s + 0.339·26-s + 0.192·27-s + 0.153·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.404904422\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.404904422\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 - 0.816T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 17 | \( 1 + 3.98T + 17T^{2} \) |
| 19 | \( 1 + 0.455T + 19T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 + 6.17T + 29T^{2} \) |
| 37 | \( 1 - 6.04T + 37T^{2} \) |
| 41 | \( 1 + 7.31T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 - 9.12T + 47T^{2} \) |
| 53 | \( 1 - 7.67T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 2.84T + 97T^{2} \) |
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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
−8.382297940751761295948518459494, −7.52308151757486138162639841773, −6.88401711131940827148652083483, −6.26337208479280772633189800900, −5.18661447352115404787823369962, −4.26737871639865541362049198979, −3.71101596378610299698878073944, −2.24257732366334340902510246583, −1.85147600246697983505702360769, −0.74378589905941118362761005257,
0.74378589905941118362761005257, 1.85147600246697983505702360769, 2.24257732366334340902510246583, 3.71101596378610299698878073944, 4.26737871639865541362049198979, 5.18661447352115404787823369962, 6.26337208479280772633189800900, 6.88401711131940827148652083483, 7.52308151757486138162639841773, 8.382297940751761295948518459494