L(s) = 1 | − 0.626·2-s + 3-s − 1.60·4-s − 1.73·5-s − 0.626·6-s − 3.52·7-s + 2.25·8-s + 9-s + 1.08·10-s − 5.10·11-s − 1.60·12-s − 6.25·13-s + 2.21·14-s − 1.73·15-s + 1.79·16-s + 17-s − 0.626·18-s − 1.32·19-s + 2.79·20-s − 3.52·21-s + 3.19·22-s − 4.67·23-s + 2.25·24-s − 1.98·25-s + 3.91·26-s + 27-s + 5.67·28-s + ⋯ |
L(s) = 1 | − 0.442·2-s + 0.577·3-s − 0.803·4-s − 0.776·5-s − 0.255·6-s − 1.33·7-s + 0.798·8-s + 0.333·9-s + 0.343·10-s − 1.53·11-s − 0.464·12-s − 1.73·13-s + 0.590·14-s − 0.448·15-s + 0.449·16-s + 0.242·17-s − 0.147·18-s − 0.303·19-s + 0.624·20-s − 0.770·21-s + 0.682·22-s − 0.974·23-s + 0.461·24-s − 0.397·25-s + 0.768·26-s + 0.192·27-s + 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.04931329400\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04931329400\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 0.626T + 2T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 + 4.67T + 23T^{2} \) |
| 29 | \( 1 + 1.94T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 + 5.78T + 41T^{2} \) |
| 43 | \( 1 - 2.16T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 + 5.16T + 59T^{2} \) |
| 61 | \( 1 + 9.98T + 61T^{2} \) |
| 67 | \( 1 + 2.06T + 67T^{2} \) |
| 71 | \( 1 - 7.20T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 4.30T + 97T^{2} \) |
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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
−8.297489366388059566888973913439, −7.76125100391119354633920398615, −7.42893686489713294921608594763, −6.36552014927312977941356356035, −5.26969703490787741654221048123, −4.64854146173712777165519200973, −3.71542140061498311194867805920, −3.04130833612495432027441425706, −2.08227058711084845438935601374, −0.12285119499056015249624317130,
0.12285119499056015249624317130, 2.08227058711084845438935601374, 3.04130833612495432027441425706, 3.71542140061498311194867805920, 4.64854146173712777165519200973, 5.26969703490787741654221048123, 6.36552014927312977941356356035, 7.42893686489713294921608594763, 7.76125100391119354633920398615, 8.297489366388059566888973913439