L(s) = 1 | − 1.41·2-s − 3-s + 5-s + 1.41·6-s − 4.82·7-s + 2.82·8-s + 9-s − 1.41·10-s − 0.414·11-s − 2.58·13-s + 6.82·14-s − 15-s − 4.00·16-s + 0.585·17-s − 1.41·18-s − 19-s + 4.82·21-s + 0.585·22-s − 2.82·24-s + 25-s + 3.65·26-s − 27-s − 4.82·29-s + 1.41·30-s + 1.82·31-s + ⋯ |
L(s) = 1 | − 1.00·2-s − 0.577·3-s + 0.447·5-s + 0.577·6-s − 1.82·7-s + 0.999·8-s + 0.333·9-s − 0.447·10-s − 0.124·11-s − 0.717·13-s + 1.82·14-s − 0.258·15-s − 1.00·16-s + 0.142·17-s − 0.333·18-s − 0.229·19-s + 1.05·21-s + 0.124·22-s − 0.577·24-s + 0.200·25-s + 0.717·26-s − 0.192·27-s − 0.896·29-s + 0.258·30-s + 0.328·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 0.585T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 4.41T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 1.34T + 79T^{2} \) |
| 83 | \( 1 - 5.89T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 - 7.89T + 97T^{2} \) |
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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
−7.51126949512983060446314822320, −6.83388125283098878425887828914, −6.25548324717876500852143584162, −5.58661093457357192122355741391, −4.72159338932858544548994028952, −3.92121973850710448654993743159, −2.96538523722708420612808991922, −2.06754996351332411220976742569, −0.831312318839799693996945119279, 0,
0.831312318839799693996945119279, 2.06754996351332411220976742569, 2.96538523722708420612808991922, 3.92121973850710448654993743159, 4.72159338932858544548994028952, 5.58661093457357192122355741391, 6.25548324717876500852143584162, 6.83388125283098878425887828914, 7.51126949512983060446314822320