L(s) = 1 | − 1.41·3-s + 1.41·5-s + 7-s + 1.00·9-s − 1.41·13-s − 2.00·15-s + 1.41·19-s − 1.41·21-s + 1.00·25-s + 1.41·35-s + 2.00·39-s + 1.41·45-s + 49-s − 2.00·57-s + 1.41·59-s + 1.41·61-s + 1.00·63-s − 2.00·65-s − 2·71-s − 1.41·75-s − 2·79-s − 0.999·81-s − 1.41·83-s − 1.41·91-s + 2.00·95-s − 1.41·101-s − 2.00·105-s + ⋯ |
L(s) = 1 | − 1.41·3-s + 1.41·5-s + 7-s + 1.00·9-s − 1.41·13-s − 2.00·15-s + 1.41·19-s − 1.41·21-s + 1.00·25-s + 1.41·35-s + 2.00·39-s + 1.41·45-s + 49-s − 2.00·57-s + 1.41·59-s + 1.41·61-s + 1.00·63-s − 2.00·65-s − 2·71-s − 1.41·75-s − 2·79-s − 0.999·81-s − 1.41·83-s − 1.41·91-s + 2.00·95-s − 1.41·101-s − 2.00·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8670724974\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8670724974\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 1.41T + T^{2} \) |
| 5 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−10.14746318908492613647259307666, −9.914221623344445245420765247528, −8.797568559389099666788830059643, −7.51603439764461457949066151747, −6.78870191631647409778969357441, −5.59548830859865741659255979477, −5.40223632646380237037549214418, −4.51621428394311017800297436680, −2.57535299837503157275197784454, −1.35654336297365492313584603706,
1.35654336297365492313584603706, 2.57535299837503157275197784454, 4.51621428394311017800297436680, 5.40223632646380237037549214418, 5.59548830859865741659255979477, 6.78870191631647409778969357441, 7.51603439764461457949066151747, 8.797568559389099666788830059643, 9.914221623344445245420765247528, 10.14746318908492613647259307666