| L(s) = 1 | + 1.41·2-s − 3-s + 1.00·4-s − 1.41·5-s − 1.41·6-s + 9-s − 2.00·10-s − 1.00·12-s + 1.41·15-s − 0.999·16-s + 1.41·17-s + 1.41·18-s − 1.41·20-s − 1.41·23-s + 1.00·25-s − 27-s + 1.41·29-s + 2.00·30-s − 1.41·32-s + 2.00·34-s + 1.00·36-s − 37-s − 1.41·45-s − 2.00·46-s + 0.999·48-s − 49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 3-s + 1.00·4-s − 1.41·5-s − 1.41·6-s + 9-s − 2.00·10-s − 1.00·12-s + 1.41·15-s − 0.999·16-s + 1.41·17-s + 1.41·18-s − 1.41·20-s − 1.41·23-s + 1.00·25-s − 27-s + 1.41·29-s + 2.00·30-s − 1.41·32-s + 2.00·34-s + 1.00·36-s − 37-s − 1.41·45-s − 2.00·46-s + 0.999·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7286629132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7286629132\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + T^{2} \) |
| 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 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
−13.86662501688781585150006845396, −12.50712601288678636476174285911, −12.08169828841517647750547962500, −11.37051472087599925796301330751, −10.09400887276507614513350913030, −8.109234277758529601641032437684, −6.88701971246819184873514860915, −5.66648124971551170411397274205, −4.53540235685744608690914638694, −3.54574592690469057076498645767,
3.54574592690469057076498645767, 4.53540235685744608690914638694, 5.66648124971551170411397274205, 6.88701971246819184873514860915, 8.109234277758529601641032437684, 10.09400887276507614513350913030, 11.37051472087599925796301330751, 12.08169828841517647750547962500, 12.50712601288678636476174285911, 13.86662501688781585150006845396
