L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 5·11-s − 14-s + 16-s − 17-s − 19-s − 20-s − 5·22-s + 25-s − 28-s + 2·29-s + 32-s − 34-s + 35-s − 7·37-s − 38-s − 40-s + 2·41-s − 8·43-s − 5·44-s + 3·47-s − 6·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 1.50·11-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.229·19-s − 0.223·20-s − 1.06·22-s + 1/5·25-s − 0.188·28-s + 0.371·29-s + 0.176·32-s − 0.171·34-s + 0.169·35-s − 1.15·37-s − 0.162·38-s − 0.158·40-s + 0.312·41-s − 1.21·43-s − 0.753·44-s + 0.437·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.96583578383507, −12.63036050190554, −12.33198332150099, −11.64188399519047, −11.33975699595312, −10.72126516876998, −10.37967606109464, −10.05304025959045, −9.307577381303705, −8.866940102126998, −8.127244844303750, −7.971324272340008, −7.377030980644970, −6.849255544487956, −6.417451569969211, −5.859775422247034, −5.330662198065231, −4.780604770957730, −4.554263756985544, −3.754879901794168, −3.143751345025061, −2.995245065725519, −2.131491515737318, −1.721016696306619, −0.6560156840949051, 0,
0.6560156840949051, 1.721016696306619, 2.131491515737318, 2.995245065725519, 3.143751345025061, 3.754879901794168, 4.554263756985544, 4.780604770957730, 5.330662198065231, 5.859775422247034, 6.417451569969211, 6.849255544487956, 7.377030980644970, 7.971324272340008, 8.127244844303750, 8.866940102126998, 9.307577381303705, 10.05304025959045, 10.37967606109464, 10.72126516876998, 11.33975699595312, 11.64188399519047, 12.33198332150099, 12.63036050190554, 12.96583578383507