L(s) = 1 | + 2-s − 1.93·3-s + 4-s + 5-s − 1.93·6-s − 0.482·7-s + 8-s + 0.732·9-s + 10-s + 1.86·11-s − 1.93·12-s − 2.35·13-s − 0.482·14-s − 1.93·15-s + 16-s + 1.44·17-s + 0.732·18-s − 3.24·19-s + 20-s + 0.931·21-s + 1.86·22-s − 1.93·24-s + 25-s − 2.35·26-s + 4.38·27-s − 0.482·28-s − 6.20·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.447·5-s − 0.788·6-s − 0.182·7-s + 0.353·8-s + 0.244·9-s + 0.316·10-s + 0.561·11-s − 0.557·12-s − 0.652·13-s − 0.128·14-s − 0.498·15-s + 0.250·16-s + 0.351·17-s + 0.172·18-s − 0.745·19-s + 0.223·20-s + 0.203·21-s + 0.397·22-s − 0.394·24-s + 0.200·25-s − 0.461·26-s + 0.843·27-s − 0.0911·28-s − 1.15·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 + 0.482T + 7T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 29 | \( 1 + 6.20T + 29T^{2} \) |
| 31 | \( 1 + 1.16T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 + 6.49T + 53T^{2} \) |
| 59 | \( 1 + 2.29T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 0.899T + 73T^{2} \) |
| 79 | \( 1 + 7.27T + 79T^{2} \) |
| 83 | \( 1 - 9.22T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.63733104447908469186570240775, −6.63538990716043114975866948637, −6.43240334436314358467968162972, −5.56646715216237301308404303130, −5.08975733387073086811195193479, −4.30674294556599271878095920435, −3.40160701087026213255971084919, −2.40688575816660916776572732934, −1.39203107513938771338666242878, 0,
1.39203107513938771338666242878, 2.40688575816660916776572732934, 3.40160701087026213255971084919, 4.30674294556599271878095920435, 5.08975733387073086811195193479, 5.56646715216237301308404303130, 6.43240334436314358467968162972, 6.63538990716043114975866948637, 7.63733104447908469186570240775