L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 7-s − 3·8-s + 9-s + 12-s + 2·13-s − 14-s − 16-s − 3·17-s + 18-s + 3·19-s + 21-s − 23-s + 3·24-s + 2·26-s − 27-s + 28-s − 6·29-s + 2·31-s + 5·32-s − 3·34-s − 36-s + 3·37-s + 3·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.688·19-s + 0.218·21-s − 0.208·23-s + 0.612·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.883·32-s − 0.514·34-s − 1/6·36-s + 0.493·37-s + 0.486·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.19799320318545522734021235065, −6.51570943080216693549911519248, −5.80233612756440293164316033462, −5.44202875618313513840059434687, −4.52675509986928312423035698646, −4.01085224716519738659409288160, −3.28993178057140937999833190571, −2.36556436780444431368022967570, −1.07447149848989966509778994328, 0,
1.07447149848989966509778994328, 2.36556436780444431368022967570, 3.28993178057140937999833190571, 4.01085224716519738659409288160, 4.52675509986928312423035698646, 5.44202875618313513840059434687, 5.80233612756440293164316033462, 6.51570943080216693549911519248, 7.19799320318545522734021235065