L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s + 9-s + 5.77·11-s − 12-s + 3.77·13-s + 3·14-s + 16-s − 0.772·17-s + 18-s + 7.77·19-s − 3·21-s + 5.77·22-s − 23-s − 24-s + 3.77·26-s − 27-s + 3·28-s + 3·29-s − 9.54·31-s + 32-s − 5.77·33-s − 0.772·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 0.333·9-s + 1.74·11-s − 0.288·12-s + 1.04·13-s + 0.801·14-s + 0.250·16-s − 0.187·17-s + 0.235·18-s + 1.78·19-s − 0.654·21-s + 1.23·22-s − 0.208·23-s − 0.204·24-s + 0.739·26-s − 0.192·27-s + 0.566·28-s + 0.557·29-s − 1.71·31-s + 0.176·32-s − 1.00·33-s − 0.132·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.518450771\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.518450771\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 0.772T + 17T^{2} \) |
| 19 | \( 1 - 7.77T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 + 6.77T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 + 7.77T + 43T^{2} \) |
| 47 | \( 1 - 8.77T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 9.54T + 61T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 + 6.54T + 73T^{2} \) |
| 79 | \( 1 - 2.22T + 79T^{2} \) |
| 83 | \( 1 - T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.624662193428648596741312374836, −7.64720658310094092628907440411, −6.95143108837804120489992740897, −6.27004389690175097510921151758, −5.45249188588146681034931797416, −4.88740666918478260601738167244, −3.92284933200266000148552037066, −3.40299506289506123473802370012, −1.75264818642675391252098622110, −1.21103198880016341753953261262,
1.21103198880016341753953261262, 1.75264818642675391252098622110, 3.40299506289506123473802370012, 3.92284933200266000148552037066, 4.88740666918478260601738167244, 5.45249188588146681034931797416, 6.27004389690175097510921151758, 6.95143108837804120489992740897, 7.64720658310094092628907440411, 8.624662193428648596741312374836