L(s) = 1 | − 2.23·2-s − 2·3-s + 3.00·4-s + 4.47·6-s + 3.23·7-s − 2.23·8-s + 9-s + 0.763·11-s − 6.00·12-s − 4.85·13-s − 7.23·14-s − 0.999·16-s + 1.61·17-s − 2.23·18-s − 6.47·21-s − 1.70·22-s − 4.47·23-s + 4.47·24-s + 10.8·26-s + 4·27-s + 9.70·28-s − 4.09·29-s − 6·31-s + 6.70·32-s − 1.52·33-s − 3.61·34-s + 3.00·36-s + ⋯ |
L(s) = 1 | − 1.58·2-s − 1.15·3-s + 1.50·4-s + 1.82·6-s + 1.22·7-s − 0.790·8-s + 0.333·9-s + 0.230·11-s − 1.73·12-s − 1.34·13-s − 1.93·14-s − 0.249·16-s + 0.392·17-s − 0.527·18-s − 1.41·21-s − 0.364·22-s − 0.932·23-s + 0.912·24-s + 2.12·26-s + 0.769·27-s + 1.83·28-s − 0.759·29-s − 1.07·31-s + 1.18·32-s − 0.265·33-s − 0.620·34-s + 0.500·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2357030908\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2357030908\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 + 7.09T + 41T^{2} \) |
| 43 | \( 1 + 8.47T + 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 + 8.47T + 67T^{2} \) |
| 71 | \( 1 - 1.23T + 71T^{2} \) |
| 73 | \( 1 + 5.56T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 8.32T + 89T^{2} \) |
| 97 | \( 1 + 0.0901T + 97T^{2} \) |
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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
−7.80669947915824863712391959216, −7.24160893893448324486650255273, −6.59047888941572921282542421178, −5.77772767210869708758353102335, −4.99896786249472197348797751905, −4.61253166921471552737494813986, −3.27653902451895067109061060331, −1.96507461885938257255111882315, −1.58402223836770699819465286519, −0.32162293321254812869381010003,
0.32162293321254812869381010003, 1.58402223836770699819465286519, 1.96507461885938257255111882315, 3.27653902451895067109061060331, 4.61253166921471552737494813986, 4.99896786249472197348797751905, 5.77772767210869708758353102335, 6.59047888941572921282542421178, 7.24160893893448324486650255273, 7.80669947915824863712391959216