L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s − 4·7-s + 8-s − 2·9-s + 3·10-s − 6·11-s + 12-s + 2·13-s − 4·14-s + 3·15-s + 16-s + 3·17-s − 2·18-s + 5·19-s + 3·20-s − 4·21-s − 6·22-s − 3·23-s + 24-s + 4·25-s + 2·26-s − 5·27-s − 4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 1.80·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 1.14·19-s + 0.670·20-s − 0.872·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s − 0.962·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.968033122\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.968033122\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 89 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.09971700535401348969711896103, −12.01521975837500572096911321539, −10.38183771626813550069949920480, −9.924983367076292701096052683568, −8.721731410942193674670586885226, −7.36410689745433254804873728837, −5.94744023022306454130283790935, −5.48197881161062206485371067956, −3.31534392384618004485979133214, −2.51994269176562419526281033451,
2.51994269176562419526281033451, 3.31534392384618004485979133214, 5.48197881161062206485371067956, 5.94744023022306454130283790935, 7.36410689745433254804873728837, 8.721731410942193674670586885226, 9.924983367076292701096052683568, 10.38183771626813550069949920480, 12.01521975837500572096911321539, 13.09971700535401348969711896103