L(s) = 1 | − 1.41·5-s + 9-s + 1.41·13-s + 1.00·25-s + 1.41·29-s + 1.41·37-s − 1.41·45-s + 49-s − 1.41·53-s − 1.41·61-s − 2.00·65-s − 2·73-s + 81-s − 2·89-s + 1.41·101-s − 1.41·109-s − 2·113-s + 1.41·117-s + ⋯ |
L(s) = 1 | − 1.41·5-s + 9-s + 1.41·13-s + 1.00·25-s + 1.41·29-s + 1.41·37-s − 1.41·45-s + 49-s − 1.41·53-s − 1.41·61-s − 2.00·65-s − 2·73-s + 81-s − 2·89-s + 1.41·101-s − 1.41·109-s − 2·113-s + 1.41·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9178624505\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9178624505\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.29410864900117724793665024404, −9.224417178210767078790269983144, −8.332701245674958334154968292582, −7.74365646789190347392103510063, −6.88828219855895510435231648978, −6.01105411178247300894644883720, −4.56928156035584895740022700438, −4.05080874669039861715738670388, −3.04750944891858721985596704686, −1.22823143032157164972006080966,
1.22823143032157164972006080966, 3.04750944891858721985596704686, 4.05080874669039861715738670388, 4.56928156035584895740022700438, 6.01105411178247300894644883720, 6.88828219855895510435231648978, 7.74365646789190347392103510063, 8.332701245674958334154968292582, 9.224417178210767078790269983144, 10.29410864900117724793665024404