L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 6.47·11-s + 4.47·13-s − 15-s − 2·17-s + 2.47·19-s − 21-s − 4·23-s + 25-s + 27-s − 2·29-s − 1.52·31-s − 6.47·33-s + 35-s − 6.94·37-s + 4.47·39-s − 2·41-s − 8.94·43-s − 45-s − 12.9·47-s + 49-s − 2·51-s − 3.52·53-s + 6.47·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s − 1.95·11-s + 1.24·13-s − 0.258·15-s − 0.485·17-s + 0.567·19-s − 0.218·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s − 0.371·29-s − 0.274·31-s − 1.12·33-s + 0.169·35-s − 1.14·37-s + 0.716·39-s − 0.312·41-s − 1.36·43-s − 0.149·45-s − 1.88·47-s + 0.142·49-s − 0.280·51-s − 0.484·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 6.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 3.52T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.720208689851675313184870434916, −8.231559789212914968370942016211, −7.54184284136052384727656407423, −6.65757471508839333768896666758, −5.64190670081187305953470451430, −4.80023078263050211009762599004, −3.64294536223962972591107727734, −3.01510831961249725461044771348, −1.82209123262909725550112808649, 0,
1.82209123262909725550112808649, 3.01510831961249725461044771348, 3.64294536223962972591107727734, 4.80023078263050211009762599004, 5.64190670081187305953470451430, 6.65757471508839333768896666758, 7.54184284136052384727656407423, 8.231559789212914968370942016211, 8.720208689851675313184870434916