L(s) = 1 | − 1.52·2-s − 2.61·3-s + 0.323·4-s + 2.08·5-s + 3.98·6-s + 2.52·7-s + 2.55·8-s + 3.82·9-s − 3.18·10-s + 11-s − 0.845·12-s + 4.21·13-s − 3.85·14-s − 5.45·15-s − 4.54·16-s − 17-s − 5.82·18-s + 4.62·19-s + 0.676·20-s − 6.60·21-s − 1.52·22-s − 2.56·23-s − 6.67·24-s − 0.638·25-s − 6.41·26-s − 2.14·27-s + 0.818·28-s + ⋯ |
L(s) = 1 | − 1.07·2-s − 1.50·3-s + 0.161·4-s + 0.934·5-s + 1.62·6-s + 0.955·7-s + 0.903·8-s + 1.27·9-s − 1.00·10-s + 0.301·11-s − 0.244·12-s + 1.16·13-s − 1.02·14-s − 1.40·15-s − 1.13·16-s − 0.242·17-s − 1.37·18-s + 1.06·19-s + 0.151·20-s − 1.44·21-s − 0.325·22-s − 0.534·23-s − 1.36·24-s − 0.127·25-s − 1.25·26-s − 0.412·27-s + 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8553864519\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8553864519\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 19 | \( 1 - 4.62T + 19T^{2} \) |
| 23 | \( 1 + 2.56T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 3.37T + 37T^{2} \) |
| 41 | \( 1 + 2.06T + 41T^{2} \) |
| 47 | \( 1 - 8.00T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 0.218T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 0.842T + 71T^{2} \) |
| 73 | \( 1 + 7.25T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 2.32T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 + 9.93T + 97T^{2} \) |
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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
−7.80260866618071448113200095901, −7.21036851445742835022918386085, −6.42120595585119923803703945410, −5.62166133859000931195132990640, −5.40105243321827809257169472500, −4.48326466160992306997762985760, −3.71221205732945630584416678090, −2.00360256605835237832198155373, −1.46381358805829974583968933259, −0.65036464625733051380382550967,
0.65036464625733051380382550967, 1.46381358805829974583968933259, 2.00360256605835237832198155373, 3.71221205732945630584416678090, 4.48326466160992306997762985760, 5.40105243321827809257169472500, 5.62166133859000931195132990640, 6.42120595585119923803703945410, 7.21036851445742835022918386085, 7.80260866618071448113200095901