L(s) = 1 | + 5-s − 5·11-s − 3·13-s − 17-s + 6·19-s − 6·23-s + 25-s − 9·29-s + 4·31-s − 2·37-s − 4·41-s − 10·43-s − 47-s + 4·53-s − 5·55-s + 8·59-s − 8·61-s − 3·65-s − 12·67-s − 8·71-s − 2·73-s + 13·79-s + 4·83-s − 85-s + 4·89-s + 6·95-s + 13·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.50·11-s − 0.832·13-s − 0.242·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s − 0.328·37-s − 0.624·41-s − 1.52·43-s − 0.145·47-s + 0.549·53-s − 0.674·55-s + 1.04·59-s − 1.02·61-s − 0.372·65-s − 1.46·67-s − 0.949·71-s − 0.234·73-s + 1.46·79-s + 0.439·83-s − 0.108·85-s + 0.423·89-s + 0.615·95-s + 1.31·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4800413114\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4800413114\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.40706041135324, −13.21520176755191, −12.29789838318906, −12.04163301676510, −11.60046304849769, −10.92409405286572, −10.39261477216374, −10.06791797147396, −9.638083467769288, −9.155299091670351, −8.496134130219262, −7.879428169523519, −7.645659034613165, −7.086890344550483, −6.495064548399342, −5.797264507179394, −5.388556779577029, −5.024758625125080, −4.438247738164951, −3.620446242782462, −3.115244882017970, −2.474261628640342, −1.999775227258540, −1.312751941293158, −0.1976565167276836,
0.1976565167276836, 1.312751941293158, 1.999775227258540, 2.474261628640342, 3.115244882017970, 3.620446242782462, 4.438247738164951, 5.024758625125080, 5.388556779577029, 5.797264507179394, 6.495064548399342, 7.086890344550483, 7.645659034613165, 7.879428169523519, 8.496134130219262, 9.155299091670351, 9.638083467769288, 10.06791797147396, 10.39261477216374, 10.92409405286572, 11.60046304849769, 12.04163301676510, 12.29789838318906, 13.21520176755191, 13.40706041135324