L(s) = 1 | + 2.80·2-s + 3-s + 5.86·4-s − 3.14·5-s + 2.80·6-s − 7-s + 10.8·8-s + 9-s − 8.81·10-s + 3.00·11-s + 5.86·12-s − 2.80·14-s − 3.14·15-s + 18.6·16-s + 3.41·17-s + 2.80·18-s − 4.86·19-s − 18.4·20-s − 21-s + 8.42·22-s + 5.37·23-s + 10.8·24-s + 4.87·25-s + 27-s − 5.86·28-s + 0.848·29-s − 8.81·30-s + ⋯ |
L(s) = 1 | + 1.98·2-s + 0.577·3-s + 2.93·4-s − 1.40·5-s + 1.14·6-s − 0.377·7-s + 3.82·8-s + 0.333·9-s − 2.78·10-s + 0.905·11-s + 1.69·12-s − 0.749·14-s − 0.811·15-s + 4.65·16-s + 0.828·17-s + 0.660·18-s − 1.11·19-s − 4.11·20-s − 0.218·21-s + 1.79·22-s + 1.12·23-s + 2.20·24-s + 0.975·25-s + 0.192·27-s − 1.10·28-s + 0.157·29-s − 1.60·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.500091166\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.500091166\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 + 4.86T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 - 0.848T + 29T^{2} \) |
| 31 | \( 1 - 0.451T + 31T^{2} \) |
| 37 | \( 1 + 3.88T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 47 | \( 1 + 9.33T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 1.45T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 + 2.36T + 71T^{2} \) |
| 73 | \( 1 - 1.24T + 73T^{2} \) |
| 79 | \( 1 + 7.10T + 79T^{2} \) |
| 83 | \( 1 - 5.04T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 4.29T + 97T^{2} \) |
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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
−8.150392606823054187177865824830, −7.62566035265847962632651564635, −6.84822892295773117092266486944, −6.37583494675996856946221175327, −5.33144495965882722642319777758, −4.41498827639895397860936689603, −3.96812966945126389828641148108, −3.32260075055847068942325343756, −2.62667884265689247968168695724, −1.31600049400441345545571918451,
1.31600049400441345545571918451, 2.62667884265689247968168695724, 3.32260075055847068942325343756, 3.96812966945126389828641148108, 4.41498827639895397860936689603, 5.33144495965882722642319777758, 6.37583494675996856946221175327, 6.84822892295773117092266486944, 7.62566035265847962632651564635, 8.150392606823054187177865824830