L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s + 3·11-s − 12-s − 13-s − 2·14-s − 15-s + 16-s + 18-s + 5·19-s + 20-s + 2·21-s + 3·22-s − 5·23-s − 24-s + 25-s − 26-s − 27-s − 2·28-s + 9·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.436·21-s + 0.639·22-s − 1.04·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.478293736\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.478293736\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.58091710524123, −13.25139123795046, −12.59400605723128, −12.10095124096956, −11.81762921528680, −11.51561612902403, −10.60783622936929, −10.22555750582169, −9.919481405911651, −9.292968615798081, −8.821050693772235, −8.083271042039982, −7.492911429366930, −6.807539906975962, −6.572675549452842, −6.112717386863427, −5.451605444361816, −5.109665872078497, −4.424734841230476, −3.837802894993350, −3.316055170504828, −2.718163665414044, −1.954390967147324, −1.297722690589686, −0.5442047617537461,
0.5442047617537461, 1.297722690589686, 1.954390967147324, 2.718163665414044, 3.316055170504828, 3.837802894993350, 4.424734841230476, 5.109665872078497, 5.451605444361816, 6.112717386863427, 6.572675549452842, 6.807539906975962, 7.492911429366930, 8.083271042039982, 8.821050693772235, 9.292968615798081, 9.919481405911651, 10.22555750582169, 10.60783622936929, 11.51561612902403, 11.81762921528680, 12.10095124096956, 12.59400605723128, 13.25139123795046, 13.58091710524123