L(s) = 1 | − 1.96·2-s + 2.86·4-s + 0.891·5-s − 3.66·8-s + 9-s − 1.75·10-s + 0.184·13-s + 4.34·16-s − 1.96·18-s + 1.47·19-s + 2.55·20-s − 0.205·25-s − 0.362·26-s − 0.547·29-s − 1.70·31-s − 4.87·32-s + 2.86·36-s − 2.90·38-s − 3.26·40-s − 0.547·43-s + 0.891·45-s + 49-s + 0.403·50-s + 0.528·52-s + 1.47·53-s + 1.07·58-s − 1.20·59-s + ⋯ |
L(s) = 1 | − 1.96·2-s + 2.86·4-s + 0.891·5-s − 3.66·8-s + 9-s − 1.75·10-s + 0.184·13-s + 4.34·16-s − 1.96·18-s + 1.47·19-s + 2.55·20-s − 0.205·25-s − 0.362·26-s − 0.547·29-s − 1.70·31-s − 4.87·32-s + 2.86·36-s − 2.90·38-s − 3.26·40-s − 0.547·43-s + 0.891·45-s + 49-s + 0.403·50-s + 0.528·52-s + 1.47·53-s + 1.07·58-s − 1.20·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5637895879\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5637895879\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 991 | \( 1 - T \) |
good | 2 | \( 1 + 1.96T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.891T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.184T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.47T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.547T + T^{2} \) |
| 31 | \( 1 + 1.70T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.547T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.47T + T^{2} \) |
| 59 | \( 1 + 1.20T + T^{2} \) |
| 61 | \( 1 + 1.70T + T^{2} \) |
| 67 | \( 1 - 1.86T + T^{2} \) |
| 71 | \( 1 + 1.20T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.184T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−9.980911131146013692540150170981, −9.395835702656542510738426773844, −8.837245918185945548508717395488, −7.59419028474705204208639563027, −7.27617277312588840028535899886, −6.23693121137785148016196430347, −5.42651688631274990259601185204, −3.48605483129506436566348788045, −2.15723917911012223970967134675, −1.30794087503013461078391524466,
1.30794087503013461078391524466, 2.15723917911012223970967134675, 3.48605483129506436566348788045, 5.42651688631274990259601185204, 6.23693121137785148016196430347, 7.27617277312588840028535899886, 7.59419028474705204208639563027, 8.837245918185945548508717395488, 9.395835702656542510738426773844, 9.980911131146013692540150170981