L(s) = 1 | − 2-s + 1.79·3-s + 4-s + 5-s − 1.79·6-s + 1.36·7-s − 8-s + 0.221·9-s − 10-s + 6.45·11-s + 1.79·12-s + 5.87·13-s − 1.36·14-s + 1.79·15-s + 16-s − 2.55·17-s − 0.221·18-s + 20-s + 2.44·21-s − 6.45·22-s + 8.42·23-s − 1.79·24-s + 25-s − 5.87·26-s − 4.98·27-s + 1.36·28-s + 4.45·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.03·3-s + 0.5·4-s + 0.447·5-s − 0.732·6-s + 0.514·7-s − 0.353·8-s + 0.0738·9-s − 0.316·10-s + 1.94·11-s + 0.518·12-s + 1.62·13-s − 0.363·14-s + 0.463·15-s + 0.250·16-s − 0.620·17-s − 0.0522·18-s + 0.223·20-s + 0.533·21-s − 1.37·22-s + 1.75·23-s − 0.366·24-s + 0.200·25-s − 1.15·26-s − 0.959·27-s + 0.257·28-s + 0.827·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.826636302\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.826636302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 - 6.45T + 11T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 23 | \( 1 - 8.42T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 + 1.40T + 31T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 + 5.48T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 - 4.42T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6.46T + 61T^{2} \) |
| 67 | \( 1 + 6.71T + 67T^{2} \) |
| 71 | \( 1 - 1.93T + 71T^{2} \) |
| 73 | \( 1 + 6.28T + 73T^{2} \) |
| 79 | \( 1 + 3.59T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 1.89T + 89T^{2} \) |
| 97 | \( 1 + 0.123T + 97T^{2} \) |
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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
−8.800666162751773333679477944603, −8.175181167665953063898296028414, −7.06617841229052733732105459445, −6.55581511918746208439626625229, −5.79295758771252942974426453388, −4.57750432691491522607031079776, −3.62875865783140298815836492239, −2.97558437239897550944020018974, −1.73954682964613463167377127745, −1.21221108375692081971758347650,
1.21221108375692081971758347650, 1.73954682964613463167377127745, 2.97558437239897550944020018974, 3.62875865783140298815836492239, 4.57750432691491522607031079776, 5.79295758771252942974426453388, 6.55581511918746208439626625229, 7.06617841229052733732105459445, 8.175181167665953063898296028414, 8.800666162751773333679477944603