L(s) = 1 | + 2-s − 3-s + 4-s − 0.573·5-s − 6-s + 7-s + 8-s + 9-s − 0.573·10-s + 2.60·11-s − 12-s + 5.61·13-s + 14-s + 0.573·15-s + 16-s + 0.560·17-s + 18-s + 6.43·19-s − 0.573·20-s − 21-s + 2.60·22-s + 5.22·23-s − 24-s − 4.67·25-s + 5.61·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.256·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.181·10-s + 0.785·11-s − 0.288·12-s + 1.55·13-s + 0.267·14-s + 0.147·15-s + 0.250·16-s + 0.135·17-s + 0.235·18-s + 1.47·19-s − 0.128·20-s − 0.218·21-s + 0.555·22-s + 1.09·23-s − 0.204·24-s − 0.934·25-s + 1.10·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.565672397\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.565672397\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
good | 5 | \( 1 + 0.573T + 5T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 0.560T + 17T^{2} \) |
| 19 | \( 1 - 6.43T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 - 5.10T + 59T^{2} \) |
| 61 | \( 1 + 1.39T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 1.42T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.69220076915240392330253736329, −6.91449983509872547901836731770, −6.41402922094206595524817505057, −5.66368531216052168102543615504, −5.09816031853412388234286425855, −4.28963117197516096331511370559, −3.63144442544584079149829513236, −2.95480369416646028220085283970, −1.56421246354036235099225584053, −0.979809292569977367186490493329,
0.979809292569977367186490493329, 1.56421246354036235099225584053, 2.95480369416646028220085283970, 3.63144442544584079149829513236, 4.28963117197516096331511370559, 5.09816031853412388234286425855, 5.66368531216052168102543615504, 6.41402922094206595524817505057, 6.91449983509872547901836731770, 7.69220076915240392330253736329