| L(s) = 1 | − 1.17·2-s − 3-s − 0.612·4-s − 5-s + 1.17·6-s − 0.648·7-s + 3.07·8-s + 9-s + 1.17·10-s − 3.99·11-s + 0.612·12-s − 13-s + 0.763·14-s + 15-s − 2.39·16-s + 3.70·17-s − 1.17·18-s − 2.56·19-s + 0.612·20-s + 0.648·21-s + 4.70·22-s + 5.59·23-s − 3.07·24-s + 25-s + 1.17·26-s − 27-s + 0.397·28-s + ⋯ |
| L(s) = 1 | − 0.832·2-s − 0.577·3-s − 0.306·4-s − 0.447·5-s + 0.480·6-s − 0.245·7-s + 1.08·8-s + 0.333·9-s + 0.372·10-s − 1.20·11-s + 0.176·12-s − 0.277·13-s + 0.204·14-s + 0.258·15-s − 0.599·16-s + 0.899·17-s − 0.277·18-s − 0.588·19-s + 0.137·20-s + 0.141·21-s + 1.00·22-s + 1.16·23-s − 0.628·24-s + 0.200·25-s + 0.230·26-s − 0.192·27-s + 0.0750·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4002472553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4002472553\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 7 | \( 1 + 0.648T + 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 5.59T + 23T^{2} \) |
| 29 | \( 1 - 0.446T + 29T^{2} \) |
| 37 | \( 1 - 0.136T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 0.971T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 - 8.06T + 53T^{2} \) |
| 59 | \( 1 + 3.19T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 4.65T + 83T^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 + 4.82T + 97T^{2} \) |
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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
−8.193394557710546723197364309684, −7.38740683762272777691197076680, −6.99084014684854756360203455398, −5.90034081576355252778511285071, −5.10454317732594661816415178662, −4.66228480195735190210878656120, −3.65151043105256264949604427212, −2.72698528223818142575483695431, −1.47114998191284908478641958958, −0.41021904730105604163777015096,
0.41021904730105604163777015096, 1.47114998191284908478641958958, 2.72698528223818142575483695431, 3.65151043105256264949604427212, 4.66228480195735190210878656120, 5.10454317732594661816415178662, 5.90034081576355252778511285071, 6.99084014684854756360203455398, 7.38740683762272777691197076680, 8.193394557710546723197364309684
