L(s) = 1 | + 2.14·2-s + 3-s + 2.60·4-s + 2.14·6-s + 2.74·7-s + 1.29·8-s + 9-s − 3.74·11-s + 2.60·12-s + 6.29·13-s + 5.89·14-s − 2.43·16-s + 2.14·18-s + 19-s + 2.74·21-s − 8.03·22-s + 0.543·23-s + 1.29·24-s + 13.4·26-s + 27-s + 7.14·28-s + 3·29-s + 1.45·31-s − 7.80·32-s − 3.74·33-s + 2.60·36-s + 5.20·37-s + ⋯ |
L(s) = 1 | + 1.51·2-s + 0.577·3-s + 1.30·4-s + 0.875·6-s + 1.03·7-s + 0.456·8-s + 0.333·9-s − 1.12·11-s + 0.750·12-s + 1.74·13-s + 1.57·14-s − 0.608·16-s + 0.505·18-s + 0.229·19-s + 0.599·21-s − 1.71·22-s + 0.113·23-s + 0.263·24-s + 2.64·26-s + 0.192·27-s + 1.35·28-s + 0.557·29-s + 0.261·31-s − 1.37·32-s − 0.652·33-s + 0.433·36-s + 0.855·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.096504289\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.096504289\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 + 3.74T + 11T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 0.543T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 1.45T + 31T^{2} \) |
| 37 | \( 1 - 5.20T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 + 8.49T + 61T^{2} \) |
| 67 | \( 1 + 2.83T + 67T^{2} \) |
| 71 | \( 1 - 7.83T + 71T^{2} \) |
| 73 | \( 1 - 7.58T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 + 2.25T + 83T^{2} \) |
| 89 | \( 1 - 4.49T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−9.532140065828462688864815531328, −8.376242849134875061007593236510, −8.080166020208417316138290174897, −6.89144635828577464043125168307, −6.01854000434960821990958855695, −5.17071929849830422259778729572, −4.50086211635991439638389892579, −3.55509853241516110056284173074, −2.75590044579045009746241961286, −1.57432478050364414297587662814,
1.57432478050364414297587662814, 2.75590044579045009746241961286, 3.55509853241516110056284173074, 4.50086211635991439638389892579, 5.17071929849830422259778729572, 6.01854000434960821990958855695, 6.89144635828577464043125168307, 8.080166020208417316138290174897, 8.376242849134875061007593236510, 9.532140065828462688864815531328